: Investigating Quadratic Functions in the Standard Form, ff(xx) = aaxx 22 + bbxx + cc Opening Exercise 1. Marshall had the equation y = (x 2) 2 + 4 and knew that he could easily find the vertex. Sarah said that he could find the y-intercept if he rewrote his equation into standard form. Sarah got him started by rewriting his equation as shown below. Continue Sarah s work to show the steps Marshall would need to take to get a y-intercept of 8 from his original equation. y = (x 2) 2 + 4 y = (x 2) (x 2) + 4 2. What is the vertex of Marshall s quadratic? 3. Use the vertex and y-intercept to sketch a graph of Marshall s parabola. What could you do to get a more accurate graph? Unit 10: Introduction to Quadratics and Their Transformations S.31
Discussion 4. A. What feature(s) of this quadratic function are visible when it is presented in the standard form, ff(xx) = aaxx 2 + bbxx + cc? B. What feature(s) of this quadratic function are visible when it is rewritten in vertex form, ff(xx) = aa(xx h) 2 + kk? 5. A general strategy for graphing a quadratic function from the standard form is: 6. Graph the function nn(xx) = xx 2 6xx + 5, and identify the key features. Unit 10: Introduction to Quadratics and Their Transformations S.32
Practice: Vertex Form to Standard Form For each exercise below, rewrite the vertex form into standard form and then identify all the important features of each quadratic. 7. Vertex form: y = (x + 6) 2 4 Key Features Vertex: y-intercept: Does the graph open up or open down? Axis of symmetry: Standard form: 8. Vertex form: y = (x 1 ) 2 7 Key Features Vertex: y-intercept: Does the graph open up or open down? Axis of symmetry: Standard form: 9. Vertex form: y = (x 3 ) 2 + 2 Key Features Vertex: y-intercept: Does the graph open up or open down? Axis of symmetry: Standard form: Unit 10: Introduction to Quadratics and Their Transformations S.33
Application Problem 10. A high school baseball player throws a ball straight up into the air for his math class. The math class was able to determine that the relationship between the height of the ball and the time since it was thrown could be modeled by the function h(tt) = 16tt 2 + 96tt + 6, where tt represents the time (in seconds) since the ball was thrown, and h represents the height (in feet) of the ball above the ground. A. The domain is and it represents. B. What does the range of this function represent? C. What is the time (t) when the ball is thrown? D. At what height does the ball get thrown? E. In vertex form the equation is h(t) = -16(t 3) 2 + 150. What is the vertex of this parabola? F. What is the maximum height that the ball reaches while in the air? How long will the ball take to reach its maximum height? G. It would be difficult to tell from the equation how many seconds it takes the ball to hit the ground. Graph the equation in the grid at the right and make an estimate. Unit 10: Introduction to Quadratics and Their Transformations S.34
Lesson Summary The standard form of a quadratic function is ff(xx) = aaxx 2 + bbxx + cc, where aa 0. From the standard form you can easily see that the y-intercept is at c. A general strategy to graphing a quadratic function from the standard form: Look for hints in the function s equation for general shape, direction, and yy-intercept. Use a T-chart to find more points on the graph. Remember that a parabola is symmetric about the vertex. This can help you identify other points you may need. Plot the points that you know (at least three are required for a unique quadratic function), sketch the graph of the curve that connects them, and identify the key features of the graph. Unit 10: Introduction to Quadratics and Their Transformations S.35
Homework Problem Set 1. Graph each quadratic equation given both the standard and vertex forms below. A. ff(xx) = xx 2 2xx 15 or ff(xx) = (xx 1) 2 16 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0-5 -4-3 -2-1 0 1 2 3 4 5-2 -3-4 -5-6 -7-8 -9-10 -11-12 -13-14 -15-16 -17-18 -19-20 B. ff(xx) = xx 2 + 2xx + 15 or ff(xx) = (xx 1) 2 + 16 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0-5 -4-3 -2-1 0 1 2 3 4 5-2 -3-4 -5-6 -7-8 -9-10 -11-12 -13-14 -15-16 -17-18 -19-20 2. The equation in Part B of Problem 1 is the product of 1 and the equation in Part A. What effect did multiplying the equation by 1 have on the graph? Unit 10: Introduction to Quadratics and Their Transformations S.36
3. Paige wants to start a summer lawn-mowing business. She comes up with the following profit function that relates the total profit to the rate she charges for a lawn-mowing job: PP(xx) = xx 2 + 40xx 100 or PP(xx) = (xx 20) 2 + 300. Both profit and her rate are measured in dollars. A. Graph the function to help you answer the following questions. B. According to the function, what is her initial cost (e.g., maintaining the mower, buying gas, advertising)? Explain your answer in the context of this problem. C. Between what two prices does she have to charge to make a profit? D. If she wants to make $275 profit this summer, is this the right business choice? Explain. 300 290 280 270 260 250 240 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0-10 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Unit 10: Introduction to Quadratics and Their Transformations S.37
4. A student throws a bag of chips to her friend. Unfortunately, her friend does not catch the chips, and the bag hits the ground. The distance from the ground (height) for the bag of chips is modeled by the function h(tt) = 16tt 2 + 32tt + 4 or h(tt) = 16(tt + 1) 2 + 20, where h is the height (distance from the ground in feet) of the chips, and tt is the number of seconds the chips are in the air. A. Graph h. 20 18 16 B. From what height are the chips being thrown? Explain how you know. 14 12 10 C. What is the maximum height the bag of chips reaches while airborne? Explain how you know. 8 6 4 2 D. About how many seconds after the bag was thrown did it hit the ground? 0-1 -0.5 0 0.5 1 1.5 2 2.5 3-2 -4 E. What is the average rate of change of height for the interval from 0 to 1 second? What does that 2 number represent in terms of the context? -6-8 F. Based on your answer to part (e), what is the average rate of change for the interval from 1.5 to 2 sec.? Unit 10: Introduction to Quadratics and Their Transformations S.38