Chapter 4 Section 10

EXAMPLE 1 Write a quadratic function in vertex form Write a quadratic function for the parabola shown. SOLUTION Use vertex form because the vertex is given. y = a(x – h) 2 + k Vertex form y = a(x – 1) 2 – 2 Substitute 1 for h and –2 for k. Use the other given point, (3, 2), to find a. 2 = a(3 – 1) 2 – 2 Substitute 3 for x and 2 for y. 2 = 4a – 2 Simplify coefficient of a. 1 = a Solve for a.

EXAMPLE 1 Write a quadratic function in vertex form A quadratic function for the parabola is y = (x – 1) 2 – 2. ANSWER

EXAMPLE 2 Write a quadratic function in intercept form Write a quadratic function for the parabola shown. SOLUTION Use intercept form because the x -intercepts are given. y = a(x – p)(x – q) Intercept form y = a(x + 1)(x – 4) Substitute –1 for p and 4 for q.

EXAMPLE 2 Write a quadratic function in intercept form Use the other given point, (3, 2), to find a. 2 = a(3 + 1)(3 – 4) Substitute 3 for x and 2 for y. 2 = –4a Simplify coefficient of a. Solve for a. 1 2 – = a A quadratic function for the parabola is 1 2 – (x + 1)(x – 4).y = ANSWER

EXAMPLE 3 Write a quadratic function in standard form Write a quadratic function in standard form for the parabola that passes through the points (–1, –3), (0, –4), and (2, 6). SOLUTION STEP 1 Substitute the coordinates of each point into y = ax 2 + bx + c to obtain the system of three linear equations shown below.

EXAMPLE 3 Write a quadratic function in standard form –3 = a(–1) 2 + b(–1) + c Substitute –1 for x and 23 for y. –3 = a – b + c Equation 1 –3 = a(0) 2 + b(0) + c Substitute 0 for x and –4 for y. –4 = c Equation 2 6 = a(2) 2 + b(2) + c Substitute 2 for x and 6 for y. 6 = 4a + 2b + c Equation 3 Rewrite the system of three equations in Step 1 as a system of two equations by substituting – 4 for c in Equations 1 and 3. STEP 2

EXAMPLE 3 Write a quadratic function in standard form a – b + c = –3 Equation 1 a – b – 4 = –3 Substitute –4 for c. a – b = 1 Revised Equation 1 4a + 2b + c = 6 Equation 3 4a + 2b – 4 = 6 Substitute –4 for c. 4a + 2b = 10 Revised Equation 3 STEP 3 Solve the system consisting of revised Equations 1 and 3. Use the elimination method.

EXAMPLE 3 Write a quadratic function in standard form a – b = 1 2a – 2b = 2 4a + 2b = 10 6a = 12 a = 2 So 2 – b = 1, which means b = 1. The solution is a = 2, b = 1, and c = –4. A quadratic function for the parabola is y = 2x 2 + x – 4. ANSWER

GUIDED PRACTICE for Examples 1, 2 and 3 Write a quadratic function whose graph has the given characteristics. 1. vertex: (4, –5) passes through: (2, –1) y = (x – 4) 2 – 5 ANSWER 2. vertex: (–3, 1) passes through: (0, –8) y =  (x + 3)2 + 1 ANSWER 3. x -intercepts: –2, 5 passes through: (6, 2) y = (x + 2)(x – 5) 1 4 ANSWER

GUIDED PRACTICE for Examples 1, 2 and 3 Write a quadratic function in standard form for the parabola that passes through the given points. 4. (–1, 5), (0, –1), (2, 11) y = 4x 2 – 2x – 1 y = x 2 + x + 3. – (–2, –1), (0, 3), (4, 1) 6. (–1, 0), (1, –2), (2, –15) y =  4x 2  x + 3 ANSWER

A pumpkin tossing contest is held each year in Morton, Illinois, where people compete to see whose catapult will send pumpkins the farthest. One catapult launches pumpkins from 25 feet above the ground at a speed of 125 feet per second. EXAMPLE 4 Solve a multi-step problem Pumpkin Tossing

EXAMPLE 4 Solve a multi-step problem The table shows the horizontal distances (in feet) the pumpkins travel when launched at different angles. Use a graphing calculator to find the best- fitting quadratic model for the data.

EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Enter the data into two lists of a graphing calculator. STEP 2 Make a scatter plot of the data. Note that the points show a parabolic trend.

EXAMPLE 4 Solve a multi-step problem STEP 3 Use the quadratic regression feature to find the best fitting quadratic model for the data. STEP 4 Check how well the model fits the data by graphing the model and the data in the same viewing window.

EXAMPLE 4 Solve a multi-step problem The best-fitting quadratic model is y = –0.261x x ANSWER

GUIDED PRACTICE for Example 4 Pumpkin Tossing 7.7. In Example 4, at what angle does the pumpkin travel the farthest? Explain how you found your answer. ANSWER About 43° ; the vertex of the graph gives what angle measure produces the greatest distance.