The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}.

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1 Use a table of values to graph each equation. State the domain and range. 1. y = x 2 + 3x + 1 x y Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity. The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}. esolutions Manual - Powered by Cognero Page 1

2 2. y = 2x 2 4x + 3 x 2x 2 4x + 3 y 3 2( 3) 2 4( 3) ( 2) 2 4( 2) ( 1) 2 4( 1) (0) 2 4(0) (1) 2 4(1) (2) 2 4(2) (3) 2 4(3) Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity. The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1}. esolutions Manual - Powered by Cognero Page 2

3 3. y = x 2 3x 3 x x 2 3x 3 y 3 ( 3) 2 3( 3) ( 2) 2 3( 2) ( 1) 2 3( 1) (0) 2 3(0) (1) 2 3(1) (2) 2 3(2) 3 13 Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity. The domain is all real numbers. The range is all real numbers less than or equal to the maximum value, or {y y 0.75}. esolutions Manual - Powered by Cognero Page 3

4 4. y = 3x 2 x + 1 x 3x 2 x + 1 y 3 3( 3) 2 ( 3) ( 2) 2 ( 2) ( 1) 2 ( 1) (0) 2 (0) (1) 2 (1) (2) 2 (2) Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity. The domain is all real numbers. The range is all real numbers less than or equal to the maximum value, or. Consider y = x 2 5x Write the equation of the axis of symmetry. To find the axis of symmetry, use the equation. For this equation, a = 1 and b = 5. So, esolutions Manual - Powered by Cognero Page 4

5 6. Find the coordinates of the vertex. Is the vertex a maximum or minimum point? The vertex goes through the line of symmetry. To find the axis of symmetry, use the equation. For this equation, a = 1 and b = 5. So, So the x-value is 2.5. Substitute this into the equation to determine the y-value. So, the vertex is (2.5, 2.25). Because a > 0, the graph opens upward and the vertex is a minimum. 7. Graph the function. x y = x 2-5x + 4 y 1 ( 1) 2 5( 1) (0) 2 5(0) (1) 2 5(1) (2) 2 5(2) (3) 2 5(3) (4) 2 5(4) Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity. esolutions Manual - Powered by Cognero Page 5

6 8. SOCCER A soccer ball is kicked from ground level with an initial upward velocity of 90 feet per second. The equation h = 16t t gives the height h of the ball after t seconds. a. What is the height of the ball after one second? b. How many seconds will it take for the ball to reach its maximum height? c. When is the height of the ball 0 feet? What do these points represent in this situation? a. To find the height of the ball after one second, substitute t = 1 into the equation and solve for h. The height is 74 feet after 1 second. b. To find the maximum, find the line of symmetry. So, the ball will reach its maximum height at seconds. c. To determine when the ball is at 0 feet, substitute h = 0 into the equation and solve for t. So, the height of the ball will be 0 feet before it is kicked at t = 0 and when it returns to the ground after the kick at t = esolutions Manual - Powered by Cognero Page 6

7 Solve each equation by graphing. If integral roots cannot be found, estimate the roots to the nearest tenth. 9. x 2 + 5x + 6 = 0 Graph the related function, f (x) = x 2 + 5x + 6. The x-intercepts of the graph appear to be at 3 and 2, so the solutions are 3 and 2. Check: esolutions Manual - Powered by Cognero Page 7

8 10. x = 6x Put the equation into standard form. Graph the related function, f (x) = x 2 + 6x + 8. The x-intercepts of the graph appear to be at 4 and 2, so the solutions are 4 and 2. Check: esolutions Manual - Powered by Cognero Page 8

9 11. x 2 + 3x 1 = 0 Graph the related function, f (x) = x 2 + 3x 1. The x-intercepts are located between 0 and 1 and between 2 and 3. Make a table using an increment of 0.1 for the x-values located between 0 and 1 and between 2 and 3. x y x y For each table, the function value that is closest to zero when the sign changes is Thus, the roots are approximately 0.4 and x 2 = 12 Put the equation into standard form. Graph the related function, f (x) = x The x-intercepts are located between 4 and 3 and between 3 and 4. Make a table using an increment of 0.1 for the x-values located between 4 and 3 and between 3 and 4. x y x y For each table, the function value that is closest to zero when the sign changes is Thus, the roots are approximately 3.5 and 3.5. esolutions Manual - Powered by Cognero Page 9

10 13. BASEBALL Juan hits a baseball. The equation h = 16t t models the height h, in feet, of the ball after t seconds. How long is the ball in the air? To determine how long the ball is in the air, find the time when h = 0. The ball will be in the air 7.5 seconds. 14. CONSTRUCTION Christopher is repairing the roof on a shed. He accidentally dropped a box of nails from a height of 14 feet. This is represented by the equation h = 16t , where h is the height in feet and t is the time in seconds. Describe how the graph is related to h = t 2. The function can be written f (x) = ax 2 + c, where a = 16 and c = 14. Since 14 > 0 and > 1, the graph of h = 16t is the graph of h = t 2 vertically compressed and shifted up 14 units. Because a is negative, the graph is also reflected across the x-axis. esolutions Manual - Powered by Cognero Page 10

11 15. PARTIES Della s parents are throwing a Sweet 16 party for her. At 10:00, a ball will slide 25 feet down a pole and light up. A function that models the drop is h = t 2 + 5t + 25, where h is height in feet of the ball after t seconds. How many seconds will it take for the ball to reach the bottom of the pole? To determine how long it will take the ball to reach the bottom of the pole, put the equation into your graphing calculator and choose a suitable window. Press 2nd, CALC, zero, and then choose a left and right bound to find when the ball reaches the bottom of the pole. From the graph we can see that the ball reaches the bottom of the pole after 8.09 seconds. Describe how the graph of each function is related to the graph of f (x) = x g(x) = x The graph of f (x) = x 2 + c represents a translation up or down of the parent graph. Since c = 3, the translation is up. So, the graph is shifted up 3 units from the parent function. esolutions Manual - Powered by Cognero Page 11

12 17. h(x) = 2x 2 The graph of f (x) = ax 2 compresses the parent graph vertically. Since a =2, the graph of y = 2x 2 is the graph of y = x 2 vertically compressed. 18. g(x) = x 2 6 The graph of f (x) = x 2 + c represents a translation up or down of the parent graph. Since c = 6, the translation is down. So, the graph is shifted down 6 units from the parent function. 19. The graph of f (x) = ax 2 represents a vertical stretch or compression of the original graph. Since a = and but, this is a vertical compression and the graph opens upward. 20. g(x) = x The function f (x) = ax 2 + c, represents a vertical stretch or compression with a vertical translation of the parent graph. Since a = 1, the graph is flipped vertically, and since c = 1, the graph is translated upward The graph of f (x) = ax 2 represents a vertical stretch or compression from the parent graph. flipped vertically. Since the graph is also compressed vertically. so the graph is esolutions Manual - Powered by Cognero Page 12

13 22. MULTIPLE CHOICE Which is an equation for the function shown in the graph? A y = 2x 2 B y = 2x C y = x 2 1 D y = 2x Since the graph opens down, the leading coefficient must be negative. The parabola is translated up 1 unit, so c = 1. Since the graph is stretched vertically, it must be of the form of f (x) = ax 2 where equation for the function is y = 2x So, the correct choice is D. Solve each equation by completing the square. Round to the nearest tenth. 23. x 2 + 4x + 2 = 0 > 1. The remaining available The solutions are about 3.4 and 0.6. esolutions Manual - Powered by Cognero Page 13

14 24. x 2 2x 10 = 0 The solutions are about 2.3 and x 2 + 4x 5 = 7 The solutions are about 3.6 and 1.6. esolutions Manual - Powered by Cognero Page 14

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