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are called exterior angles. • Angles 4 and 5 are interior angles on opposite sides of the transversal and do not have the same vertex. They are called alternate interior angles. Angles 3 and 6 are another pair of alternate interior angles. Angles and Parallel Lines 259 • Angles 1 and 8 are exterior angles on opposite ... |
and 6 are alternate interior angles of parallel lines, and alternate interior angles of parallel lines are congruent). (4) Therefore m2 m6 (because the measure of each angle is x). 3 4 5 6 2 3 6 260 Geometric Figures, Areas, and Volumes The four steps on page 259 serve as a proof of the following theorem: Theorem. If ... |
the sum of the measures of the interior angles on the same side of the transversal is 180°. EXAMPLE 1 In the figure, the parallel lines are cut by a transversal. If ml (5x 10) and m2 (3x 60), find the measures of 1 and 2. Solution Since the lines are parallel, the alternate interior angles, 1 and 2, have equal measure... |
interior angles are congruent. 21. If two parallel lines are cut by a transversal, then the alternate interior angles are comple- mentary. 22. If two parallel lines are cut by a transversal, then the corresponding angles are supplemen- tary. 23. In the figure on the right, two parallel lines are cut by a transversal. ... |
of the triangle that form the right, are called the legs of the right triangle. The side opposite the GH angle, right angle, and GI HI, is called the hypotenuse. Sum of the Measures of the Angles of a Triangle When we change the shape of a triangle, changes take place also in the measures of its angles. Is there any r... |
x 3x C x A x 2x 3x 180 6x 180 x 30 2x 60 3x 90 Check 60 2(30) 90 3(30) 30 60 90 180 ✔ Answer mA 30, mB 60, mC 90 Classifying Triangles According to Sides T Z N L Scalene triangle M R S Isosceles triangle X Equilateral triangle Y • A scalene triangle has no sides equal in length. • An isosceles triangle has two sides eq... |
BC, mB mA. Therefore, mA mB mC. In an equilateral triangle, the measures of all of the angles are equal. In DEF, all of the angles are equal in measure. Since mD mE, EF DF; also, since mD mF, EF DE. Therefore, DE EF DF, and DEF is equilateral. If a triangle is equilateral, then it is equiangular. Properties of Special... |
5. 30°, 110°, 40° In 6–9, find, in each case, the measure of the third angle of the triangle if the measures of two angles are: 6. 60°, 40° 7. 100°, 20° 8. 54.5°, 82.3° 9. 241 48, 811 28 268 Geometric Figures, Areas, and Volumes 10. What is the measure of each angle of an equiangular triangle? 11. Can a triangle have:... |
. The measure of the vertex angle of an isosceles triangle is 3 times the measure of each base angle. Find the number of degrees in each angle of the triangle. 22. The measure of the vertex angle of an isosceles triangle is 15° more than the measure of each base angle. Find the number of degrees in each angle of the tr... |
mz. b. What kind of a triangle is EFG? 32. In RST, mR x, mS x 30, mT x 30. a. Find the measures of the three angles of the triangle. b. What kind of a triangle is RST? 33. In KLM, mK 2x, mL x 30, mM 3x 30. a. Find the measures of the three angles of the triangle. b. What kind of a triangle is KLM? G z x 130° E A y 140... |
ray R and the point of intersection on the other ray T. STEP 4. Using the same opening of the compass as was used in step 3, place the point of the compass at M and draw an arc that intersects the ray and extends beyond the ray. (Draw at least half of a circle.) Label the point of intersection L. STEP 5. Place the poi... |
will learn how to construct an angle bisector. An angle bisector is the line that divides an angle into two congruent angles. STEP 1. Use the straightedge to draw an acute angle. Label the vertex S. STEP 2. With any convenient opening of the compass, place the point at S and draw an arc that intersects both rays of S.... |
D and A. Two angles that are not consecutive angles are called opposite angles; A and C are opposite angles, and B and D are opposite angles. AB Special Quadrilaterals When we vary the shape of the quadrilateral by making some of its sides parallel, by making some of its sides equal in length, or by making its angles ... |
Diagonal ADC. AC divides quadrilateral ABCD into two triangles, ABC and (2) The sum of the measures of the angles of ABC is 180°, and the sum of the measures of the angles of ADC is 180°. (3) The sum of the measures of all the angles of ABC and ADC together A B is 360°. (4) Therefore, mA mB mC mD 360. The Family of Pa... |
A and D are interior angles on the same side of a transversal, and these angles have been shown to be supplementary. (3) In parallelogram ABCD, AD and BC ments of parallel lines cut by transversal are segAB. D 180 – x C x x A 180 – x B (4) Since mA x, mB 180 x because A and B are interior angles on the same side of a ... |
pezoids A trapezoid has one and only one pair of parallel lines. Each of the two parallel sides is called a base of the trapezoid. Therefore, we can use what we know about angles formed when parallel lines are cut by a transversal to demonstrate some facts about the angles of a trapezoid. Quadrilateral ABCD is a trapez... |
angles then the parallelogram is a rectangle. Do you agree with Adam? Explain why or why not. 2. Emmanuel said that if a parallelogram has one right angle then the parallelogram is a rec- tangle. Do you agree with Emmanuel? Explain why or why not. Developing Skills In 3–8, in each case, is the statement true or false?... |
divided into rectangle ABFG, isosceles trapezoid BCEF, and isosceles triangle CDE, as shown in the diagram. a. Find the sum of the measures of the interior angles of ABCDEFG by using the sum of the measures of the angles of the two quadrilaterals and the triangle. b. Sketch the heptagon on your answer paper and show h... |
three right angles. 7-6 AREAS OF IRREGULAR POLYGONS Many polygons are irregular figures for which there is no formula for the area. However, the area of such figures can often be found by separating the figure into regions with known area formulas and adding or subtracting these areas to find the required area. EXAMPL... |
operations, make sure to keep at least one extra digit for intermediate calculations to avoid roundoff error. Alternatively, when working with a calculator, you can round at the end of the entire calculation. EXERCISES Writing About Mathematics 1. ABCD is a trapezoid with AB i CD. The area of ABD is 35 square units. W... |
. ABCD is a trapezoid with BC'AB BE 5.00 cm, and EC 7.00 cm. Find the area of triangle AED. and E a point on BC. AB 12.0 cm, DC = 24.0 cm, 282 Geometric Figures, Areas, and Volumes D C 15 yd 13 yd 18 yd B E 38 yd Ex. 9 A D 6.0 mm E 4.0 mm C 8.0 mm A Ex. 10 F 5.0 mm B 9. ABCD is a quadrilateral with DE'AB BC 13 yd, and ... |
right prism The surface area of a solid is the sum of the areas of the surfaces that make up the solid. The surface area of a right prism is the sum of the areas of the bases and the faces of the solid. The number of faces of a prism is equal to the number of sides of a base. Surface Areas of Solids 283 • If the base ... |
the same size, and with a height that is perpendicular to these bases. Fruits and vegetables are often purchased in “cans” that are in the shape of a right circular cylinder. The label on the can corresponds to the surface area of the curved portion of the cylinder. That label is a rectangle whose length is the circum... |
regular hexagons. The measure of each of the six sides of the hexagon is represented by a and the height of the solid by 2a. a. How many surfaces make up the solid? b. Describe the shape of each face c. Express the dimensions and the area of each face in terms of a. 2. A regular hexagon can be divided into six equilat... |
Geometric Figures, Areas, and Volumes 12. Sandi wants to make a pillow in the shape of a right circular cylinder. The diameter of the circular ends is 10.0 inches and the length of the pillow (the height of the cylinder) is 16.0 inches. Find the number of square inches of fabric Sandi needs to make the pillow to the c... |
There are 3 layers, each containing 8 cubes, for a total of 24 cubes. Note that 3 corresponds to the height, h, that 8 corresponds to the area of the base, B, and that 24 corresponds to the volume V in cubic units. A cube that measures 1 foot on each side represents 1 cubic foot. Each face of the cube is 1 square foot... |
third the volume of a right prism with the same base and same height as those of the pyramid. The volume of a cone is onethird of the volume of a right circular cylinder, when both the cone and the cylinder have circular bases and heights that are equal in measure. The formula for the volume of the cone is ) h ( t h g ... |
. Answer The volume of ice cream is 200 cubic centimeters. Error in Geometric Calculations When a linear measure is used to find area or volume, any error in the linear value will be increased in the higher-dimension calculations. EXAMPLE 3 The length of a side of a cube that is actually 10.0 centimeters is measured to... |
. Each pie is approximately a right circular cylinder. Compare the volume of one 8-inch pie to that of one 4-inch pie. 2. Tennis balls can be purchased in a cylindrical can in which three balls are stacked one above the other. How does the radius of each ball compare with the height of the can in which they are package... |
of 15 meters. a. Express the volume of the cylinder in terms of p. b. Find the volume of the cylinder to the nearest ten cubic meters. 15. The base of a cone has a radius of 7 inches. The height of the cone is 5 inches. a. Find the volume of the cone in terms of p. b. Find the volume of the cone to the nearest cubic i... |
soda the can holds. In 24–26, express each answer to the correct number of significant digits. 24. Cynthia used a shipping carton that is a rectangular solid measuring 12.0 inches by 15.0 inches by 3.20 inches. What is the volume of the carton? 25. The highway department stores sand in piles that are approximately the... |
An equilateral triangle is equiangular. The sum of the measures of the angles of a quadrilateral is 360°. The sum of the measures of the angles of any polygon with n sides is 180(n 2). If the measures of the bases of a trapezoid are b1 and b2 and the measure of 1 2h(b11b2). the altitude is h, then the formula for the ... |
• Bases of a trapezoid • Parallelogram • Rectangle • Rhombus • Square • Isosceles trapezoid 7-7 Right prism • Face • Surface area • Rectangular solid • Right circular cylinder 7-8 Volume • Pyramid • Cone • Sphere • Center of a sphere • Radius of a sphere • Diameter of a sphere REVIEW EXERCISES 1. If g AB g CD find mAE... |
the measures of the angles? 14. The measure of the smaller of two supplementary angles is of the mea- 4 5 sure of the larger. What are the measures of the angles? 15. The vertices of a trapezoid are A(3, 1), B(7, 1), C(5, 5), and D(7, 5). a. Draw ABCD on graph paper. b. E is the point on CD such that CD'AE. What are t... |
centimeters of water does it hold? c. How many liters of water does the trough hold? (1 liter 1,000 cm3) Exploration A sector of a circle is a fractional part of the interior of a circle, determined by an angle whose vertex is at the center of the circle (a central angle). The area of a sector depends on the measure o... |
0.09 " (2) 0.9 " 4. The product (a2b)(a3b) is equivalent to (3) 2–3 (4) 0.15 (1) a6b 5. The solution set of (1) {24} (2) a5b 3x 1 7 5 1 2 (2) {24} 6x 2 5 6. The sum of b2 7 and b2 3b is (3) a5b2 (4) a6b2 is (3) {6} (4) {6} (1) b4 4b (2) b4 3b 7 (3) 2b2 4b (4) 2b2 3b 7 7. Two angles are supplementary. If the measure of... |
two packages. The larger package weighed 12 ounces more than the smaller. If the total weight of the packages was 17 pounds, how much did each package weigh? 14. a. Solve for x in terms of a and b: ax 3b 7. b. Find, to the nearest hundredth, the value of x when a b. 5 " Part IV and 3 " Answer all questions in this par... |
onometric functions have applications beyond the study of triangles, in this chapter we will limit the applications to the study of right triangles. 8-1 THE PYTHAGOREAN THEOREM The Pythagorean Theorem 301 The solutions of many problems require the measurement of line segments and angles. When we use a ruler or tape mea... |
area can also be expressed as 302 Trigonometry of the Right Triangle Area of the square area of the four triangles area of the smaller square 4 1 2 ab c2 A Although the area is written in two different ways, both expressions are equal. B Thus, (a b)2 4 1 2 ab c2. A If we simplify, we obtain the relationship of the Pyt... |
¯ 130 ENTER DISPLAY Therefore, to the nearest tenth, the length of the hypotenuse is 11.4. Note that the calculator gives only the positive rational approximation of the square root of 130. EXAMPLE 1 A ladder is placed 5 feet from the foot of a wall. The top of the ladder reaches a point 12 feet above the ground. Find... |
a right triangle, the longest side, whose measure is 8, must be the hypotenuse. Then: 82 c2 a2 b2 72 1 42 5? 5? 64 64 65 ✘ 49 1 16 Answer The triangle is not a right triangle. The Pythagorean Theorem 305 EXERCISES Writing About Mathematics 1. A Pythagorean triple is a set of three positive integers that make the equat... |
by 40 centimeters 24. 28 feet by 45 feet 25. 17 meters by 144 meters 26. 15 yards by 20 yards 27. 18 millimeters by 24 millimeters 306 Trigonometry of the Right Triangle 28. The diagonal of a rectangle measures 65 centimeters. The length of the rectangle is 33 cen- timeters. Consider the measurements to be exact. a. F... |
distance from home plate to first base is 90.0 feet, approximate, to the nearest tenth of a foot, the distance from home plate to second base. 8-2 THE TANGENT RATIO Naming Sides The Tangent Ratio 307 In a right triangle, the hypotenuse, which is the longest side, is opposite the right angle. The other two sides in a r... |
, with mC 90, the definition of the tangent of A is as follows: tangent A length of side opposite /A length of side adjacent to /A BC AC 5 a b By using “tan A” as an abbreviation for tangent A, “opp” as an abbreviation for the length of the leg opposite A, and “adj” as an abbreviation for the length of the leg adjacent... |
, enter the sequence of keys shown below. ENTER: TAN 60 ) ENTER DISPLAY, The value given in the calculator display is the rational approximation of " the value of tan 60° that we found using the ratio of the lengths of the legs of a right triangle with a 60° angle. Therefore, to the nearest ten-thousandth, tan 60° 1.73... |
angle is 1. 2. Use one of the right triangles formed by drawing an altitude of an equilateral triangle to find tan 30°. Express the answer that you find to the nearest hundred-thousandth and compare this result to the valued obtained from a calculator. Developing Skills In 3–6, find: a. tan A b. tan B 3. √18 4. B 3 90... |
In rectangle ABCD, AB 10 and BC 5. a. Find tan CAB. b. Find the measure of CAB to the nearest degree. c. Find tan CAD. d. Find the measure of CAD to the nearest degree. 34. In ABC, C is a right angle, mA = 45, AC = 4, BC = 4, and AB a. Using the given lengths, write the ratio for tan A. 4 " 2. b. Use a calculator to f... |
measured from B to O is congruent to the angle of depression measured from O to B. g HO g BA g OB 7 Using the Tangent Ratio to Solve Problems Procedure To solve a problem by using the tangent ratio: 1. For the given problem, make a diagram that includes a right triangle. Label the known measures of the sides and angle... |
ve for x. Answer To the nearest meter, the height of the tree is 35 meters. Applications of the Tangent Ratio 315 EXAMPLE 3 From the top of a lighthouse 165 feet above sea level, the measure of the angle of depression of a boat at sea is 35.0°. Find to the nearest foot the distance from the boat to the foot of the ligh... |
° 13 ft 10 ft 50 ft 55° x 6.0 ft 9.0 ft x 10. 60° x 68° x 20 ft 5. 40° 18 ft x 8. 12 ft 24 ft x 11. 8.0 ft x 8.0 ft Applying Skills 12. At a point on the ground 52 meters from the foot of a tree, the measure of the angle of ele- vation of the top of the tree is 48°. Find the height of the tree to the nearest meter. 13.... |
other and bisect each other. In BD rhombus ABCD, diagonals sure of each angle to the nearest degree. a. mBCM b. mMBC and AC meet at M. If BD 14 and AC 20, find the mea- c. mABC d. mBCD 8-4 THE SINE AND COSINE RATIOS Since the tangent is the ratio of the lengths of the two legs of a right triangle, it is not directly u... |
b c By using “cos A” as an abbreviation for cosine A, “adj” as an abbreviation for the length of the leg adjacent to A, and “hyp” as an abbreviation for the length of the hypotenuse, we can shorten the way we write the definition of cosine A as follows: B a c A b C cos A adj hyp AC AB 5 b c The Sine and Cosine Ratios ... |
C e. Use a calculator. Start with the ratio in part c and use SIN1 or start with the ratio in part d and use COS1. METHOD 1 sin B = 24 25 METHOD 2 cos B = 7 25 ENTER: 2nd SIN1 24 25 ENTER: 2nd COS1 7 25 ) ENTER ) ENTER DISPLAY ) DISPLAY mB 74 to the nearest degree. Answer EXERCISES Writing About Mathematics 1. If A an... |
cos A 0.8695 39. a. Use a calculator to find sin 25° and sin 50°. b. If the measure of an angle is doubled, is the sine of the angle also doubled? 40. a. Use a calculator to find cos 25° and cos 50°. b. If the measure of an angle is doubled, is the cosine of the angle also doubled? 41. As an angle increases in measure... |
D c. sin CBD d. cos CBD 50. In right triangle ABC, C is the right angle, BC 1, AC 3 " a. Using the given lengths, write the ratios for sin A and cos A. and AB 2. b. Use a calculator to find sin 30° and cos 30°. c. What differences, if any, exist between the answers to parts a and b? 51. In ABC, mC 90 and sin A cos A. F... |
K 90 38.0 52.0. METHOD 1 METHOD 2 sin B sin 38.0° KG BK x 322 cos K cos 52.0° KG BK x 322 x 322 sin 38.0° x 198.242995 x 322 cos 52.0° x 198.242995 Write the ratio: Substitute the given values: Solve for x: Compute using a calculator: Answer The height of the kite to the nearest foot is 198 feet. 324 Trigonometry of th... |
Writing About Mathematics 1. Brittany said that for all acute angles, A, (tan A)(cos A) sin A. Do you agree with Brittany? Explain why or why not. 2. Pearl said that as the measure of an acute angle increases from 1° to 89°, the sine of the angle increases and the cosine of the angle decreases. Therefore, cos A is the... |
to the top. Find to the nearest degree the measure of the angle that the road makes with the horizontal. 22. A 25-foot ladder leans against a building and makes an angle of 72° with the ground. Find to the nearest foot the distance between the foot of the ladder and the building. 23. A wire 2.4 meters in length is att... |
of a house that is 21 feet above the ground. The measure of the steepest angle that a ladder can make with the house when it is placed directly under the roof is 27°. Find the length of the shortest ladder that can be used to reach the roof, to the nearest foot. 8-6 SOLVING PROBLEMS USING TRIGONOMETRIC RATIOS When the... |
the measures of two sides of a right triangle are given, it is possible to find the measures of the third side and of the acute angles. Explain how you would find these measures. 2. If the measures of the acute angles of a right triangle are given, is it possible to find the measures of the sides? Explain why or why n... |
altitude is 6. Find to the nearest degree the measure of the smaller acute angle of ABC. 330 Trigonometry of the Right Triangle 20. In ABC, AB = 30, mB = 42, mC 36, and AD is an altitude. a. Find to the nearest integer the length of AD. b. Using the result of part a, find to the nearest integer the length of DC. 21. A... |
angle of 11° with the ground. Find to the nearest hundred feet the distance the airplane has traveled when it has attained an altitude of 450 feet. 30. Find to the nearest degree the measure of the angle of elevation of the sun if a child 88 cen- timeters tall casts a shadow 180 centimeters long. 31. CD and AB represe... |
line of sight to an object at a lower elevation. Angle of depression F D E G Angle of elevation VOCABULARY 8-1 Trigonometry • Direct measurement • Indirect measurement • Pythagorean Theorem • Pythagorean triple 8-2 Opposite side • Adjacent side • Similar • Tangent of an acute angle of a right triangle 8-3 Angle of ele... |
BC 5, and AC 9. Find to the nearest degree the measure of A. 19. Find to the nearest meter the height of a building if its shadow is 42 meters long when the angle of elevation of the sun measures 42°. 20. A 5-foot wire attached to the top of a tent pole reaches a stake in the ground 3 feet from the foot of the pole. F... |
, in terms of n and s, the measure of the altitude to the base of one of the isosceles triangles. d. Express the area of one of the isosceles triangles in terms of n and s. e. Write a formula for the area of a regular polygon in terms of the measure of a side, s, and the number of sides, n. CUMULATIVE REVIEW CHAPTERS 1... |
00 centimeters. The surface area of the cylinder to the correct number of significant digits is (1) 1,610 square centimeters (2) 1,620 square centimeters (3) 1,860 square centimeters (4) 1,870 square centimeters 9. When 5b2 2b is subtracted from 8b the difference is (3) 5b2 10b (2) 5b2 6b (1) 6b 5b2 (4) 5b2 6b 10. When... |
not play basketball. What are the career plans of each boy and what sport does he play? Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part... |
Functions Using Their Slopes 9-8 Graphing Direct Variation 9-9 Graphing First-Degree Inequalities in Two Variables 9-10 Graphs Involving Absolute Value 9-11 Graphs Involving Exponential Functions Chapter Summary Vocabulary Review Exercises Cumulative Review 337 338 Graphing Linear Functions and Relations 9-1 SETS, REL... |
is not between 1 and 3. EXAMPLE 1 List the elements of each set or indicate that the set is the empty set. a. A {x x of the set of natural numbers and x 0} c. C {x x 3 5} b. B {2n n of the set of whole numbers} d. D {m m is a multiple of 5 and m 25} Sets, Relations, and Functions 339 Solution a. Since there are no nat... |
airs The relation can be listed in set notation as shown below: {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)} 340 Graphing Linear Functions and Relations 3. Table of Values The ordered pairs shown in 2 can be displayed in a table. Graph In the coordinate plane, the domain is a subset of the numbers on the x-axis and... |
x-coordinate) and y with the second element of the pair (the y-coordinate). If the result is a true statement, the ordered pair is a solution of the equation. • (1.5, 13) {(x, y) y 2x 10} because 13 2(1.5) 10 is true. • (4, 14) {(x, y) y 2x 10} because 14 2(4) 10 is false. Since in the equation y 2x 10, x represents t... |
2) Choose any five values for x. Since no replacement set is given, any real numbers can be used: (3) For each selected value of x, determine y: 3x 3x y 7 3x y 3x 7 x 2 0 1 3 3 5 3x 7 3(2) 7 3(0) 7 A 1 1 7 23 3 B 3(3) 7 3(5) 7 y 13 7 6 2 8 Answer (–2, 13), (0, 7),, (3, –2), (5, –8) 1 3, 6 A B Note that many other solut... |
(1) For the pair (155,?), x 155 and y is to be determined. Then: y 64.00 0.25x y 64.00 0.25x 64.00 0.25(155) 64.00 38.75 102.75 When the car was driven 155 miles, the rental cost was $102.75. (2) For the pair (?, 69), y 69 and x is to be determined. Then: y 64 0.25x 69 64 0.25x 64 64 5 0.25x 0.25x 0.25 5 0.25 20 x Whe... |
w are both positive} Answer Note: Since the length and the width cannot be zero or negative, the rule must specify that the dimensions, l and w, of the pool are both positive. EXERCISES Writing About Mathematics 1. A function is a set of ordered pairs in which no two different pairs have the same first element. In Exa... |
In 28–31, state which sets of ordered pairs represent functions. If the set is not a function, explain why. 28. {(1, 2), (2, 3), (3, 4), (5, 6)} 30. {(5, 5), (5, 5), (6, 6), (6, 6)} 29. {(1, 2), (2, 1), (3, 4), (4, 3)} 31. {(81, 9), (81, 9), (25, 5), (25, 5)} In 32–35, find the range of each function when the domain i... |
function and determine its domain. Length (x) Width Area (A) 1 2 176 2(1) 176 2(2) 1[176 2(1)] 174 2[176 2(2)] 344 41. (1) At the Riverside Amusement Park, rides are paid for with tokens purchased at a central booth. Some rides require two tokens, and others one token. Tomas bought 10 tokens and spent them all on x tw... |
for both x and y is the set of real numbers, the following is true: All of the points whose coordinates are solutions of x y 6 lie on this same straight line, and all of the points whose coordinates are not solutions of x y 6 do not lie on this line2 –1 –1 – This line, which is the set of all the points and only the p... |
2. EXAMPLE 3 a. Write the following verbal sentence as an equation: The sum of twice the x-coordinate of a point and the y-coordinate of that point is 4. b. Graph the equation written in part a. Solution a. Let x the x-coordinate of the point, and y the y-coordinate of the point. Then: 2x y 4 Answer b. How to Proceed ... |
member of the solution set of the equation 2x y 6. 3. x 4, y 2 4. x 0, y 6 5. x 4, y 2 For 6–9, state in each case whether the point whose coordinates are given is on the graph of the given equation. 6. x y 7; (4, 3) 8. 3x 2y 8; (2, 1) 7. 2y x 7; (1, 3) 9. 2y 3x 5; (1, 4) In 10–13, find in each case the number or numb... |
37. x y 8 41. x – 2y 0 45. x 3y 12 49. 2x y 4 34. y 2x 1 38. x y 5 42. y 3x –5 46. 2x 3y 6 50. 4x 3y b. Write the coordinates of two other points on the line drawn in part a. 52. a. Through points (2, 3) and (1, 3), draw a straight line. b. Does point (0, 1) lie on the line drawn in part a? Applying Skills In 53–60: a... |
defines a function and can be displayed on a graphing calculator. If there are other equations entered in the before Y= menu, press entering the new equation. DISPLAY: ENTER: CLEAR 4 –3 –2 –1 –1 –2 –3 Y 2 GRAPH The equation of a line parallel to the x-axis and b units from the x-axis is y b. If b is positive, the line... |
0 10. y 0 15. y 2.5 6. x 3 11. y 4 x 5 23 2 16. 18. Write an equation of the line that is parallel to the x-axis 7. x 5 12. y 7 17. y 3.5 a. 1 unit above the axis. b. 5 units above the axis. c. 4 units below the axis. d. 8 units below the axis. e. 2.5 units above the axis. f. 3.5 units below the axis. 354 Graphing Lin... |
and a vertical line that separate ABCD into four congruent rectangles. d. Write the equations of the lines drawn in part c. 25. During the years 2004, 2005, and 2006, there were 5,432 AnyClothes stores. a. Write a linear equation that gives the number of stores, y. b. Predict the number of stores for the years 2007 an... |
x 356 Graphing Linear Functions and Relations When we compute the slope of a line that is determined by two points, it does not matter which point is considered as the first point and which the second. Also, when we find the slope of a line using two points on the line, it does not matter which two points on the line ... |
values of these points to the change or difference of the corresponding x-values. Thus: slope of a line difference iny-values difference in x-values Therefore: slope of g P1P2 m 2 y y 1 2 x2 2 x1 x1, can The difference in x-values, x2 be represented by x, read as “delta x.” Similarly, the difference in y-values, y1, ca... |
D. As the x-values increase, the y-values are unchanged. Between point C and point D, the change in y is 0 and the change in x is 3. Since y is 0 and x is 3, the slope of must be 0. Thus: g GH y 1 O –1 –1 G (–2, –2) C x H (1, –2) D slope of g GH m Dy Dx 5 0 3 5 0 Principle 3. If a line is parallel to the x-axis, its s... |
when y changes by 3, then x changes by 2. Start at point and move 3 units upward and 2 units to the right to locate point B: A(2, 21) (3) Start at B and repeat these movements to locate point C: (4) Draw a line that passes through points A, B, and C: C 2 B 3 2 1 –1–1 3 O 1 A(2, –1) x 360 Graphing Linear Functions and ... |
2 2 x1 2 x2 find the answer to Example 1 to justify your response. The Slope of a Line 361 2. Explain why any two points that have the same x-coordinate lie on a line that has no slope. Developing Skills In 3–8: a. Tell whether each line has a positive slope, a negative slope, a slope of zero, or no slope. b. Find the ... |
D(1, 4) are the vertices of a quadrilateral. a. Graph the points and draw quadrilateral ABCD. b. What type of quadrilateral does ABCD appear to be? c. Compute the slope of and the slope of d. What is true of the slope of and the slope of g BC g BC g AD. g AD? e. If two segments such as AD and BC, or two lines such as ... |
1 has been drawn with one leg coinciding with the x-axis and with its hypotenuse coinciding with. Similarly, right triangle 2 has been drawn with one leg coinciding with the x-axis and with its hypotenuse coinciding with /1 /2. Since triangles 1 and 2 are right triangles, tan a oppa adja tan b oppb adjb y oppa a adja ... |
21. <1 is m1, the slope of <2 is m2, and <1'<2, then m1 2 1 m2 or The following statement is also true: If the slope of <1'<2. then <1 is m1, the slope of <2 is m2, and m1 2 1 m2 or m1? m2 5 21, EXAMPLE 1 The equation of g AB is y 2x 1. a. Find the coordinates of any two points on g AB. b. What is the slope of g AB? c... |
2x 6 6. 3y x 9. x 2y 1 Applying Skills 12. A taxi driver charges $3.00 plus $0.75 per mile. a. What is the cost of a trip of 4 miles? b. What is the cost of a trip of 8 miles? c. Use the information from parts a and b to draw the graph of a linear function described by the taxi fares. d. Write an equation for y, the c... |
graph of 2x y 6 intersects the x-axis at (3, 0) and the x-intercept is 3. Divide each term of the equation 2x y 6 by the constant term, 6 y 6 5 6 2x Note that x is divided by the x-intercept, 3, and y is divided by the y-intercept, –6. The Intercepts of a Line 367 In general, if a linear equation is written in the for... |
hing Linear Functions and Relations The equation we have been studying can be compared to a general equation of the same type: y 2x 3 Q slope a y-intercept y mx b Q slope a y-intercept The following statement is true for the general equation: If the equation of a line is written in the form y mx b, then the slope of th... |
x 4 y x 3 5 1 2 1 17. 6. y x 14. 10. y 3x 7 2x 1 3 4 5 1 1 3y 4x 5 2 2y 5 1 15. 4x 3y 0 19. What do the graphs of the equations y 4x, y 4x 2, and y 4x 2 all have in 16. 2x 5y 10 18. common? 20. What do the lines that are the graphs of the equations y 2x + 1, y 3x + 1, and y 5 24x 1 1 all have in common? 21. If two line... |
(5) Use the slope to find two more points on the line. Since Dy Dx 5 22 slope, when y changes 3 by 2, x changes by 3. Therefore, start at point A and move 2 units down and 3 units to the right to locate point B. Then start at point B and repeat the procedure to locate point C: (6) Draw the line that passes through the... |
through the three points. This is the graph of 3x 2y 9: 22y 5 23x 1 9 22 1 9 y 5 23x 22 2x 2 9 y 5 3 2 2(1 26 2 y 5 23 y 1 O C B x A(1, –3) Note that the graph intersects the x-axis at B(3, 0). The x-intercept is 3. 372 Graphing Linear Functions and Relations Translating, Reflecting, and Scaling Graphs of Linear Funct... |
by a factor of 4: Then, translate the resulting function 1 unit down: y x y x 3 Answer y y 1 10x 1 10x 1 1 Answer y 4x y 4x 1 Answer EXERCISES Writing About Mathematics 1. Explain why (0, b) is always a point on the graph of y mx b. 2. Gunther said that in the first example in this section, since the slope of the line... |
(3, 5) whose slope is. 3 b. What appears to be the y-intercept of this line? c. Use the slope of the line and the answer to part b to write an equation of the line. d. Do the coordinates of point (3, 5) satisfy the equation written in part c? 28. a. Is (1, 1) a point on the graph of 3x 2y 1? b. What is the slope of 3x... |
through the origin. 1 5 3 b 5 0 The unit of measure for the lemonade and for the frozen concentrate is the same, namely, cups. Therefore, no unit of measure is associated with the ratio which, in this case, is or 3. or 6 cups 2 cups 5 3 1 lemonade concentrate 5 lemonade concentrate 5 9 cups 3 cups 5 3 1 There are many... |
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