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NationalTutoringObservatory
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MathEd-PII

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student
Hi, i need help with this specific problem on my AP precalc work
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volunteer
I would be happy to help you with that.
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student
thank u!
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volunteer
Would you like to start with a, or is there a specific one you would like clarification on?
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student
honestly the whole problem is not maaking sense bc theres no number, but i can try letter a and you can check?
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volunteer
Sure, that sounds great!
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student
alr is a, [a,b]
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volunteer
Exactly, that is correct!
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volunteer
As we move from a to b, the slope is increasing, so f(x) is concave up over that interval.
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volunteer
Do you want to try b?
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student
wait can intervals be represented in letters?
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volunteer
Yes, in this case, instead of giving you specific x values, they are giving you letters to make the problem more general.
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student
ahhh got uuu
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volunteer
Usually you would use numbers though
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student
okay ill try b
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volunteer
Also, one minor thing, when we are referring to concavity, we typically use parentheses instead of brackets
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student
okkay
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volunteer
Because nothing is happening to the slope when we are AT a or b, only the values in between
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student
alr b is (c,d)
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volunteer
Almost! f(x) is concave down over that interval, but is there anywhere else where it is concave down as well?
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student
the ending of B?
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volunteer
Yes, it is also concave down over (b, c)
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volunteer
Can you explain why that is?
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student
cuz the graph starts the shift
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student
in the direction its moving
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student
like where i circles
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volunteer
Yes, the slope starts decreasing at b
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volunteer
At b we have a slightly positive slope, and as we move towards c it gets closer to zero
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student
correct
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volunteer
And then after c it is still concave down because it keeps decreasing from 0 to a negative number
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volunteer
So what would be the interval for part b?
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student
(b,d)
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student
or infinity?
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volunteer
Great work!
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volunteer
I would go with (b, d) because we don't know what is happening after x = d
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student
true true
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student
alr lemme try c
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student
ok does it even increase at all
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volunteer
Yes, there is an interval where the graph increases. Remember, now we are talking about the graph f(x) increasing, not the slope increasing
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student
ok so i think im lost can you guide me to it so itll help me for d?
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volunteer
Of course no problem!
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volunteer
Is it ok if I erase the drawings from the picture?
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student
ofccc
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volunteer
Ok, lets start by looking at each section of the graph one by one.
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student
kk
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volunteer
Can you circle the point on f(x) where x = a?
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volunteer
Yup, that's the x value x = a; can you circle the point on the f(x) curve? In other words, the point at f(a)
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student
ooopsss
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volunteer
No problem haha
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volunteer
Can you do the same for the point on the f(x) curve where x = b
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volunteer
Great! Looking at the two circled points, which one is greater (higher on the graph/higher y-coordinate)?
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student
b ofc
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volunteer
Right! So did the f(x) have to increase or decrease to get there?
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student
increase!
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volunteer
Yes, so (a, b) is the first increasing interval!
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volunteer
Can you find any other increasing intervals?
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student
ok imma search
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student
(b,c)
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volunteer
Awesome! So what would be the final answer for c?
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student
(a,b) (b,c)
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volunteer
Yes, and we can turn those into one interval in this case since it is increasing over all of (a, c)
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student
Emily
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[ { "pii_type": "PERSON", "surrogate": "Emily", "start": 0, "end": 5 } ]
student
alr let me find outt
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student
the decreasing
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volunteer
Alright!
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student
(c,d)
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volunteer
Nice work! Also one more thing, does the worksheet give anymore information about f(x)?
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volunteer
For example does it say it is a polynomial or anything else?
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student
nope, shocking
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volunteer
Ok, I just wanted to make sure that there are no assumptions about the end behavior going to infinity or -infinity
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volunteer
Would you like to try part e?
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student
Emily
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[ { "pii_type": "PERSON", "surrogate": "Emily", "start": 0, "end": 5 } ]
student
would it be 1
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volunteer
Not quite, the zeroes of a function refers to the points where that function equals 0. In other words, f(x), the y value on this graph, is equal to zero at what x coordinate?
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student
from (b,c)
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student
?
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student
wait no
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volunteer
Unlike the other questions, zeroes are at invidual points rather than intervals.
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volunteer
Which axis would the zero of a function intersect?
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student
ok umm x axis?
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volunteer
Yes! Where does f(x) intersect the x-axis?
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student
well at 0,0
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volunteer
The x axis is the horizontal axis here. Can you circle the point where f(x) instersects that line (the x-axis)?
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student
is that right?
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volunteer
Nice try! f(x) refers to the curve. where do the curve and the horizontal line (the x-axis) intersect?
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student
oh bumsickkless
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student
that/
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volunteer
There we go, nice job! 🎉
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volunteer
So that would be the zero on this graph, because it is the only place that f(x) = 0.
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volunteer
Can you write the coordinate for that point?
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student
it don't have numbers:(
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volunteer
We can apply the same strategy we used for intervals! Instead of using numbers, we can use the letters they give us. What is the letter they give at the circled x value
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student
d
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volunteer
Yes, so we know our coordinate is (d, ?).
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student
0?
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volunteer
Awesome! The coordinate is (d, 0).
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volunteer
Would you like to try another similar problem to practice?
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student
sure but for the y intercept lemme find it
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volunteer
Ok!
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student
is that the y intercept
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