Lesson 20: Graphing Quadratic Functions

Transcription

1 Opening Exercise 1. The science class created a ball launcher that could accommodate a heavy ball. They moved the launcher to the roof of a 23-story building and launched an 8.8-pound shot put straight up into the air. (Note: Olympic and high school women use the 8.8-pound shot put in track and field competitions.) The motion is described by the function h(tt) = 16tt tt + 240, where h(tt) represents the height, in feet, of the shot put above the ground with respect to time tt in seconds. (Important: No one was harmed during this experiment!) A. Graph the function, and identify the key features of the graph. B. After how many seconds does the shot put hit the ground? C. What is the maximum height of the shot put? D. What is the value of h(0), and what does it mean for this problem? Ball Launcher Data Unit 11: More with Quadratics Factored Form S.183

2 2. Matching Match each graph with the correct equation. Be prepared to share how you know the equation and graph are a match. A. i. 1 y= ( x 3)( x+ 1) 2 B. ii. 5 y= ( x 3)( x+ 1) 4 C. iii. 1 y= ( x 3)( x+ 1) 4 D. iv. 5 y= ( x 3)( x+ 1) 4 E. v. 1 y= ( x 3)( x+ 1) 4 Unit 11: More with Quadratics Factored Form S.184

3 3. Discussion What information did you use to match the graph to its equation? 4. Solve the following equation. xx 2 + 6xx 40 = Consider the equation yy = xx 2 + 6xx 40. A. Given this quadratic equation, find the point(s) where the graph crosses the xx-axis B. Earlier in this unit, we learned about the symmetrical nature of the graph of a quadratic function. How can we use that information to find the vertex for the graph? C. How could we find the yy-intercept (where the graph crosses the yy-axis and where xx = 0)? D. What else can we say about the graph based on our knowledge of the symmetrical nature of the graph of a quadratic function? Can we determine the coordinates of any other points? E. Plot the points you know for this equation on the grid above, and connect them to show the graph of the equation Unit 11: More with Quadratics Factored Form S.185

4 Practice Exercises Graph the following functions, and identify key features of the graph. 6. ff(xx) = (xx + 2)(xx 5) 7. gg(xx) = xx 2 5xx 24 Unit 11: More with Quadratics Factored Form S.186

5 8. ff(xx) = 5(xx 2)(xx 3) 9. pp(xx) = 6xx xx 60 Unit 11: More with Quadratics Factored Form S.187

6 10. Consider the graph of the quadratic function at the right with xx-intercepts 4 and 2. A. Write a formula for a possible quadratic function, in factored form, that the graph represents using aa as a constant factor. B. The yy intercept of the graph is 16. Use the yy-intercept to adjust your function by finding the constant factor aa. 11. Given the xx-intercepts for the graph of a quadratic function, write a possible equation for the quadratic function, in factored form. A. xx-intercepts: 0 and 3 B. xx-intercepts: 1 and 1 C. xx-intercepts: 5 and 10 D. xx-intercepts: 1 2 and 4 Unit 11: More with Quadratics Factored Form S.188

7 12. Consider the graph of the quadratic function shown at the right with xx-intercept 2. A. Write a formula for a possible quadratic function, in factored form, that the graph represents using aa as a constant factor. B. The yy-intercept of the graph is 4. Use the yy-intercept to adjust your function by finding the constant factor aa. Unit 11: More with Quadratics Factored Form S.189

8 Lesson Summary When we have a quadratic function in factored form, we can find its xx-intercepts, yyintercept, axis of symmetry, and vertex. For any quadratic equation, the roots are the solution(s) where yy = 0, and these solutions correspond to the points where the graph of the equation crosses the xx-axis. A quadratic equation can be written in the form yy = aa(xx mm)(xx nn), where mm and nn are the roots of the function. Since the xx-value of the vertex is the average of the xx-values of the two roots, we can substitute that value back into the equation to find the yy-value of the vertex. If we set xx = 0, we can find the yy-intercept. Unit 11: More with Quadratics Factored Form S.190

9 Homework Problem Set Graph the following and identify the key features of the graph. 1. ff(xx) = (xx 2)(xx + 7) 2. h(xx) = 3(xx 2)(xx + 2) Unit 11: More with Quadratics Factored Form S.191

10 3. gg(xx) = 2(xx 2)(xx + 7) 4. h(xx) = xx 2 16 Unit 11: More with Quadratics Factored Form S.192

11 5. pp(xx) = xx 2 2xx qq(xx) = 4xx xx + 24 Unit 11: More with Quadratics Factored Form S.193

12 7. A rocket is launched from a cliff. The relationship between the height of the rocket, h, in feet, and the time since its launch, tt, in seconds, can be represented by the following function: h(tt) = 16tt tt A. Sketch the graph of the motion of the rocket. B. When does the rocket hit the ground? C. When does the rocket reach its maximum height? D. What is the maximum height the rocket reaches? E. At what height was the rocket launched? 8. Given the xx-intercepts for the graph of a quadratic function, write a possible formula for the quadratic function, in factored form A. xx-intercepts: 1 and 6 B. xx-intercepts: 2 and 2 3 C. xx-intercepts: 3 and 0 D. xx-intercept: 7 Unit 11: More with Quadratics Factored Form S.194

13 9. Suppose a quadratic function is such that its graph has xx-intercepts of 3 and 2 and a yy-intercept of 6. A. Write a formula for the quadratic function. B. Sketch the graph of the function. Unit 11: More with Quadratics Factored Form S.195

14 Unit 11: More with Quadratics Factored Form S.196