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: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | that 's how they have set-up the data right over here . which statements are true about the above matrix ? select all that apply . | whats the difference between a regular matrix and a transposed matrix ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | : the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . | just a general question , but can matrices be considered as a kind of spreadsheet ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | : the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . | where do matrices appear in real life ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | you ca n't just randomly order this around . now you could represent it in other ways , you could have another ... let 's say i have the matrix a , we could have picked to do something like this where we could have said well maybe this is the toothpaste column and maybe this is the toilet paper column , and that this f... | how could a computer programmer use matrixes , while coding ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | : the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . | how the concept of matrices came & their applications ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | in duluth a toothpaste cost $ 1.95 a tube , so this data right over here , is right over here . this looks like the toothpaste row . toothpaste actually is also tp so i 'll just write out tooth . | is a matrix kind of organized like a table ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | select all that apply . the following matrix can also be used to contain the same information as g. that 's what 's interesting about the matrix , what we have right over here is essentially an encapsulation of all of the data that we have in this upper paragraph and it 's useful because a computer could make use of th... | do gpus use matrices as well for forming geometry and texture in games , programs , etc ... ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | select all that apply . the following matrix can also be used to contain the same information as g. that 's what 's interesting about the matrix , what we have right over here is essentially an encapsulation of all of the data that we have in this upper paragraph and it 's useful because a computer could make use of th... | 0.00 why did sal use matg and mato ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | you could have done something like this toothpaste in new york city was 5.25 , toilet paper in new york city is 8.95 , toothpaste in duluth is 1.95 and toilet paper in duluth is 3.99 . this would have been , this matrix a that i 've just constructed , this does contain the same data because if we appropriately define o... | how do you define what rows and columns represent ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | this is the toothpaste row , this is toothpaste in duluth , $ 1.95 . toothpaste in new york city , 5.25 . that 's how they have set-up the data right over here . | in the first question could we have taken the duluth to be a slanting line instead of a vertical line and similarly for new york city ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | select all that apply . the following matrix can also be used to contain the same information as g. that 's what 's interesting about the matrix , what we have right over here is essentially an encapsulation of all of the data that we have in this upper paragraph and it 's useful because a computer could make use of th... | can the data be perceived only as rows and columns or it can also be taken as slanting lines ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | this one is not representing the same information as that up there . you ca n't just randomly order this around . now you could represent it in other ways , you could have another ... let 's say i have the matrix a , we could have picked to do something like this where we could have said well maybe this is the toothpas... | technically , ca n't scrambled matrices ( some entries switched around ) serve as a stealthy cryptograhic method , and still be able to contain data about the values of goods in different cities , as long as you know the pattern ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | that 's how they have set-up the data right over here . which statements are true about the above matrix ? select all that apply . | is a matrix basically a chart ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | that 's how they have set-up the data right over here . which statements are true about the above matrix ? select all that apply . | what is the correct way to solve the bug matrix in the practice questions ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | toothpaste actually is also tp so i 'll just write out tooth . this is the toothpaste row , this is toothpaste in duluth , $ 1.95 . toothpaste in new york city , 5.25 . | does the subscript for g2,1 matter or is it just a typo ? |
: the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . in duluth , toothpaste costs $ 1.95 a tube while in new york city it costs $ 5.25 a tube . the data for this can be encoded in t... | : the price of things at two supermarkets are different in different cities . toilet paper in duluth , minnesota cost 3.99 a package while toilet paper in new york city cost 8.95 a package . | is there a matg '' or matg '' ' ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | now , you could probably figure out what the area of this rectangle is . it 's actually a square , because it has the same length and the same width . we multiply the length , 3 meters , times the width , so times 3 meters , to get 3 times 3 is 9 -- 9 square meters . | what is the different between width and length ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so 1 , 2 , 3 , 4 , 5 -- and you see it 's going in the right direction -- 6 , 7 , and 8 . so the area of this rectangle is , indeed , 8 square meters . and you could view this as 4 groups of 2 . | how can i mesure a rectangle that is 3d ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | we figure out how many square meters can we cover this thing with , without overlapping , without going over the boundaries . we get the exact same thing as if we multiplied 3 times 3 , if we multiplied the length times the width in meters . | does an oval have an area ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | i 've got three rectangles here , and i also have their dimensions . i have their height and their width . | what is a timestamp and how can i reference it ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | it 's actually a square , because it has the same length and the same width . we multiply the length , 3 meters , times the width , so times 3 meters , to get 3 times 3 is 9 -- 9 square meters . and let 's verify it again just to feel really good about this multiplying the dimensions of these rectangles . | can the volume resemble a 3-d space ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | how do you find area per shape ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | how do we know the area of a circle ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so it matches up . we figure out how many square meters can we cover this thing with , without overlapping , without going over the boundaries . we get the exact same thing as if we multiplied 3 times 3 , if we multiplied the length times the width in meters . | so the `` area '' is actually meaning how many unit suqares can cover this space ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . | when we were multiplying the last figure , when the height and width are the same , is that why people call some numbers `` perfect squares '' ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | what is the area of the star ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of meters , since all of the dimensions are in meters , i 'm going to measure the area in terms of square meters . so let 's see how many square meters i can fit onto this yellow rectangle without going outs... | how do you measure open figures ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . | do you know the width of a your house ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | and you might say , hey , wait . is this just a coincidence that if i took the length and i multiplied it by the width , that i get the same thing as its area ? and no , it 's not , because when you took the length , you essentially said , well , how many rows do i have ? | can you do the area divided by the length and width ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | what is the formula to find the radius , diameter , area and circumference of a circle with the radius of 12 ' ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | do all shapes have an area ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | do all shapes have an area and perimeter ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so 1 , 2 , 3 , 4 , 5 -- and you see it 's going in the right direction -- 6 , 7 , and 8 . so the area of this rectangle is , indeed , 8 square meters . and you could view this as 4 groups of 2 . | how can you measure a 3d shape , like square or rectangle ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | what is a good way to remember the difference between area and perimeter ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | can you explain to me why area is considered important here , and how to find area of rectangles well ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | well , that 's because width here is 3 meters . so i could put 3 square meters side by side . and how did i get the 2 groups ? | if a patio has a are of 25 square feet and i only want to put a railing up on 3 sides how much wood will i need ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | i want to draw it a little bit cleaner . now , you could probably figure out what the area of this rectangle is . it 's actually a square , because it has the same length and the same width . | when we 're doing this , how do you find the area if one of the perimeters is a variable , and you have to figure out the variable first ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | and let 's verify it again just to feel really good about this multiplying the dimensions of these rectangles . so we have 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , and 9 . so it matches up . | if we are given 6 large squares & 4 small squares along with a certain number of rectangles , how do i figure out the rectangular configurations ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | i want to draw it a little bit cleaner . now , you could probably figure out what the area of this rectangle is . it 's actually a square , because it has the same length and the same width . we multiply the length , 3 meters , times the width , so times 3 meters , to get 3 times 3 is 9 -- 9 square meters . and let 's ... | if you can figure out the area of a shape by multiplying the length times width , how would you figure out the area of a shape with 1 square unit times either the length or the width ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so let 's see how many square meters i can fit onto this yellow rectangle without going outside of its boundary and without overlapping . so i can fit 1 square meter . remember , a square meter is just a square where its length is 1 meter and its width is 1 meter . so that 's 1 square meter , 2 , 3 , 4 , or 5 , and 6 s... | 1 meter equals to how many inches ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so this is really a quick way of counting how many of these square meters you have . so you could say that 2 meters multiplied by 3 meters is equal to 6 square meters . now , you might say , hey , i 'm not sure if that always applies . | is there a sign for meters ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | so we see here that the area is 6 square meters . area is equal to 6 square meters . but something might be jumping out at you . | why do open figures have no area ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | and how did i get the 2 groups ? well , this has a length of 2 meters . so another way that i could have essentially counted these six things is i could have said , look , i have a length of 2 meters . | is n't length always measured as the horizontal distance ? |
i 've got three rectangles here , and i also have their dimensions . i have their height and their width . and in fact , this one right here has the same height and width , so this is actually a square . so let 's think about how much space they each take up on my screen . and since we 're doing everything in terms of ... | you might have recognized that i could view this as really 2 groups of 3 . and let me make that very clear . so , for example , i could view this as one group of 3 and then another group of 3 . | how do you make the same block reappear ? |
pascale ricketts has invented a game called three rolls to 10 . you roll a fair six-sided die three times . if the sum of the rolls is 10 or greater , you win . if it is less than 10 , you lose . what is the probability of winning three rolls to 10 ? so , there is several ways that you can approach this . the way we '... | so , based on just these 10 experiments we got a pretty clean 50 % . so , do you think the theoretical probability is actually 50 % ? maybe you 'd want to continue running these experiments over and over . | how can we calculate the theoretical probability in this and the previous examples ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it would go all the way right over here . so if i reflect a just across the y-axis , it would go there . and then if i reflected that point across the x-axis , then i would end up at 5 below the x-axis at an x-coordinate of 6 . | when would you ever need to reflect points in real world applications ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | i thought when the reflection of a y-axis is needed , you put the points vertical direction and not horizontally and vice versa for the x-axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so if i reflect a just across the y-axis , it would go there . and then if i reflected that point across the x-axis , then i would end up at 5 below the x-axis at an x-coordinate of 6 . so to go from a to b , you could reflect across the y and then the x , or you could reflect across the x , and it would get you right ... | what happens to the x and y coordinates if you reflect along line y= -x ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . let 's check our answer . | which planet/star is said to disappear fist ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so let 's think about this right over here . this is at the point negative 6 comma 5 . this is at the point negative 5 comma 6 . | how do you know which negative part to point the dot ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . | how come the guy speaks so low ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | negative 6 comma negative 7 is right there . and we are reflecting across the x-axis . so , once again , if you imagine that this is some type of a lake , or maybe some type of an upside-down lake , or a mirror , where would we think we see its reflection ? | is reflecting over both axis the same as reflection over the origin ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | how do you reflect a coordinate like ( 0,9 ) over the x-axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | how do you reflect a coordinate like ( 7,0 ) over the y-axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so we 've plotted negative 8 comma 5 . now we have to plot its reflection across the y-axis . and so you can imagine if this was some type of lake or something and you were to see its reflection , and this is , say , like the moon , you would see its reflection roughly around here . | can you use reflection to predict data trends across the other quadrants if you only had data points in q1 ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | if you were on a 4d coordinate plane would the axis be w ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | is there a z axis which would be on a 3 dimensional coordinate plane ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | negative 6 comma negative 7 is right there . and we are reflecting across the x-axis . so , once again , if you imagine that this is some type of a lake , or maybe some type of an upside-down lake , or a mirror , where would we think we see its reflection ? | why is it called a reflection across the x-axis if the y value changes ( and vice-versa ) ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | negative 6 comma negative 7 is right there . and we are reflecting across the x-axis . so , once again , if you imagine that this is some type of a lake , or maybe some type of an upside-down lake , or a mirror , where would we think we see its reflection ? | how does reflecting work if there is a z axis on your coordinate plane ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so you would see it at 8 to the right of the y-axis , which would be at positive 8 , and still 5 above the x-axis . so that 's its reflection right over here . it 's reflection is the point 8 comma 5 . | what is the idea of reflection used for outside of this exercise ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | if i were to reflect this point across the y-axis , it would go all the way to positive 6 , 5 . so it would go all the way right over here . so if i reflect a just across the y-axis , it would go there . and then if i reflected that point across the x-axis , then i would end up at 5 below the x-axis at an x-coordinate ... | why is it that when we invert across the y-axis we go left and right ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | how do you reflect across the axis x=1 ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | dose the x axis have 4 sides and the y axis have 1 side ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | how do you reflect ( 0 , -8 ) across the y-axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | negative 6 comma negative 7 is right there . and we are reflecting across the x-axis . so , once again , if you imagine that this is some type of a lake , or maybe some type of an upside-down lake , or a mirror , where would we think we see its reflection ? | i 'm still confused on to reflecting the x-axis and the y-axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | to end , on the third question , when sal says that you have to reflect across one axis , then the other , are you basically switching the numbers between the x axis and the y axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so , once again , if you imagine that this is some type of a lake , or maybe some type of an upside-down lake , or a mirror , where would we think we see its reflection ? well , its reflection would be the same distance . we 're reflecting across the x-axis , so it would be the same distance , but now above the x-axis ... | what is the distance between ( -3 , -4 ) and ( -3 , 4 ) ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i 'll go up 5 . so the y-coordinate is 5 right over here . | why is reflecting points in the coordinate plane is important ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | the point negative 6 comma negative 7 is reflec -- this should say `` reflected '' across the x-axis . plot negative 6 comma negative 7 and its reflection across the x-axis . so negative 6 comma negative 7 , so we 're going to go 6 to the left of the origin , and we 're going to go down 7 . | why in the negative side on the y axis is positive numbers and not negative ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | let 's do one more . the point b is a reflection of point a across which axis ? so let 's think about this right over here . | determine the location of point a , after a reflection a = a ' , where was point a ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it would go all the way right over here . so if i reflect a just across the y-axis , it would go there . and then if i reflected that point across the x-axis , then i would end up at 5 below the x-axis at an x-coordinate of 6 . so to go from a to b , you could reflect across the y and then the x , or you could refle... | how do you reflect multiple points across the line y=x and make it into a graph ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | and then if i reflected that point across the x-axis , then i would end up at 5 below the x-axis at an x-coordinate of 6 . so to go from a to b , you could reflect across the y and then the x , or you could reflect across the x , and it would get you right over here . it would get you to negative 6 comma 5 , and then r... | could you reflect the point diagonally ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it would go all the way right over here . so if i reflect a just across the y-axis , it would go there . and then if i reflected that point across the x-axis , then i would end up at 5 below the x-axis at an x-coordinate of 6 . | how would you reflect ( 0 , -2 ) ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | let 's do one more . the point b is a reflection of point a across which axis ? so let 's think about this right over here . | what is the point of the reflection ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | let 's see . it does n't look like it 's only one axis . if i were to reflect this point across the y-axis , it would go all the way to positive 6 , 5 . | why ca n't they be any other letters ' like `` a , '' `` b , '' or `` c '' ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so there you have it right over here . we reflected this point to right up here , because we reflected across the x-axis . let 's check our answer . | what are the coordinates of the point when ( - 3,1 ) is reflected across the x-axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | let 's do one more . the point b is a reflection of point a across which axis ? so let 's think about this right over here . | can we find a reflection point for z in a domain d = { r < z < 1 : re z > 0 , im z > 0 } and reflection across { 0 < re z < 1 , im z= 0 } axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | so it 's really reflecting across both axes . so we would reflect across the x-axis and then the y-axis . it would have also been legitimate if we said the y-axis and then the x-axis . | is there such thing of a z axis ? |
the point negative 8 comma , 5 is reflected across the y-axis . plot negative 8 comma 5 and its reflection across the y-axis . so first let 's plot negative 8 comma 5 . so its x-coordinate is negative 8 , so i 'll just use this one right over here . so the x-coordinate is negative 8 , and the y-coordinate is 5 , so i '... | let 's do one more . the point b is a reflection of point a across which axis ? so let 's think about this right over here . | in the last example can we say that point a is the reflection of z and x and y axis ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | what are the main differences between a tangent line and a secant line in a curve ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and just as a reminder , we can figure out the slope between two points . two points define a line . and between those two points , we can find the rate of change of y , with respect to x . | does a secant line always only touch 2 points on the graph ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | what is the difference between a chord and a secant ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | but using just the tools from algebra , we could at least start to think about , what is the average rate of change over the interval from x1 to x2 ? well , what 's the average rate of change ? well , that 's just how much did my y change -- so that 's my change in y -- for this change in x . | how to find out the average rate of change of an arbitrary function ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | then what is different between tangent line and secant line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | how can it be called `` secant line '' ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | well , we figured out the slope of the line that connects these two points . and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . | the curve is obviously part of a circle 's circumference ; is n't the line only tangent to the circumference ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | we can at least figure out the average rate of change of a curve , or a function , over an interval . that is the same exact thing as the slope of a secant line . now just as a little bit of foreshadowing , where is this all going ? | is there any way to use the slope of a secant line to determine the coordinates of a point on a graph where the slope of tangent line is equal to sole of the secant line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | the important thing is that you 're consistent . if you 're subtracting you 're starting value from your ending value in the numerator , you have to subtract your starting value from your ending value in the denominator as well . so this right here you probably remember from algebra class . | why ca n't the value of h be exactly equal to zero ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | or a change in y divided by a change in x . just as a reminder , this triangle , that 's the greek letter delta . it 's shorthand for change in whatever . | what does the delta symbol mean ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and let 's say we have another point all the way over here . and let 's say that this x value , this x value right over here is x sub 1 . and the y value over here is y sub 1 . | how come on the second graph , sal did n't use x naught and x 1 , but used x 1 and x 2 ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | what is the difference between a secant line and a tangent line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so let 's put two points on here . so let 's say that this point right over here , this x value , is x sub -- well , this is pronounced x naught , or x sub 0 is just x naught -- and when x is x0 for this line , y is y0 . so this is a point x0 , comma y0 . | if i wanted to find the instantaneous slope of , say , x = 3 , why could n't i just take the secant slope of 2 < = x < = 4 ( or any interval where the beginning and the end are an equal x distance away from the point in question ) ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and let 's say we have another point all the way over here . and let 's say that this x value , this x value right over here is x sub 1 . and the y value over here is y sub 1 . | if the slope is lower than it is at x=3 in the first half of the interval , and higher than x = 3 in the second half , would n't averaging the whole thing give the value of the slope at x = 3 ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | well if we 're ending here and we started here , let 's just do ending point minus starting point . so it is x1 minus x0 . and that way , doing it this way , i would have made sure that i have a positive value . | why does sal uses x1 and x2 instead of x0 and x1 , is there a change in concept after switching to a curve line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and if this had an even higher inclination like this , if it increased even more as x increased , then it would even have a higher slope . so this right over here is some line . so that 's some line . and just as a reminder , we can figure out the slope between two points . | why is a line straight ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and if this had an even higher inclination like this , if it increased even more as x increased , then it would even have a higher slope . so this right over here is some line . so that 's some line . and just as a reminder , we can figure out the slope between two points . | why ca n't a line be curved ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so the big idea here is we 're extending the idea of slope . we said , ok , we already knew how to find the slope of a line . a curve , we do n't have the tools yet , but calculus is about to give it to us . | how do you find the slope of the line ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | absolutely . you could have done that . then you would have just gotten the negative of each of these values in the numerator and denominator , but they would have canceled out . | what could i study to solidify the connection between equations and graphical representations of them and visa versa ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | and what we call a line that intersects a curve in exactly two places ? well , we figured out , we call that a secant line . so this right over here is a secant line . so the big idea here is we 're extending the idea of slope . | what is the difference between `` secant line '' and `` secant the trig function '' ? |
so let 's review the idea of slope , which you might remember from your algebra classes . the slope is just the rate of change of a line . or the rate of change of y , with respect to x , as we go along a line . and you could also view it as a measure of the inclination of a line . so the more incline the line is , the... | so let 's put two points on here . so let 's say that this point right over here , this x value , is x sub -- well , this is pronounced x naught , or x sub 0 is just x naught -- and when x is x0 for this line , y is y0 . so this is a point x0 , comma y0 . | is n't the slope of a line just y/x ? |
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