Quadratic Functions & Baseball Height

1. 5-Minute Check on Activity 4-1 Which direction does y = 3x2 open? Which function’s graph is narrower, f(x) =4x2 or g(x) = ½x2? Solve the following equations for x 8 = 2x2 27 = 3x2 y = 15 and y = 3x2 up f(x) 4 = x24 = x ± 2 = x 9 = x29 = x ± 3 = x 15 = 3x25 = x2 ± 5 = x Click the mouse button or press the Space Bar to display the answers.

2. Activity 4 - 2 Baseball and the Sears Tower

3. Objectives • Identify functions of the form y = ax² + bx + c as quadratic functions • Explore the role of a as it relates to the graph of y = ax² + bx + c • Explore the role of b as it relates to the graph of y = ax² + bx + c • Explore the role of c as it relates to the graph of y = ax² + bx + c • Note: a ≠ 0 in objectives above

4. Vocabulary • Quadratic term – the term, ax², in the quadratic equation; determines the opening direction and steepness of the curve • Linear term – the term, bx, in the quadratic equation; helps determine the turning point • Constant term – the term, c, in the quadratic equation; also graphically the y-intercept • Coefficients –the numerical factors of the quadratic and linear terms (a and b) • Turning point –the maximum or minimum location on the parabola; where it turns back

5. Activity Imagine yourself standing on the roof of the 1450-foot-high Sears Tower in Chicago. When you release and drop a baseball from the roof of the tower, the ball’s height above the ground, H (in feet), can be modeled as a function of the time (in seconds), since it was dropped. This height function is defined by: H(t) = -16t²+ 1450 acceleration constant due to gravity Height offset

6. Activity Continued Complete the table to the right: How far does the ball fall in thefirst second? How far does it fall during the2nd second? What is the average rate of change of H with respect to tin the first second? During the 2nd second? 1450 – 1434 = 16 feet 1434– 1386 = 16 feet 16 feet / sec 48 feet / sec

7. Activity Continued When does the ball hit the ground? What is the practical domainof the height function? What is the practical range ofthe height function? Now graph the function using the table to the right About 9.5 seconds 0 ≤ t ≤ 9.5 seconds 0 ≤ H ≤ 1450 feet

8. H t Activity Continued Is the shape of the curve the path of the ball? 1400 1200 1000 800 600 400 200 No, the ball falls straight down

9. Quadratic Function • Standard form: y = ax² + bx + c • Quadratic term: ax² • Determines Directiona > 0 then parabola opens upa < 0 then parabola opens down • Determines Width: The bigger |a|, the narrower the graph • Linear term: bx • If b = 0, then turning point on y-axis • If b ≠ 0, then turning point not on y-axis • Constant term: c • y-intercept is at (0, c)

10. y x The Effects of a in y = ax² + bx + c Graph the following quadratic functions: • f(x) = x² • g(x) = ½x² • h(x) = 2x² • j(x) = -2x²

11. y x The Effects of b in y = ax² + bx + c Graph the following quadratic functions: • f(x) = x² • g(x) = x² - 4x • h(x) = x² + 6x • j(x) = -x² + 6x

12. y x The Effects of c in y = ax² + bx + c Graph the following quadratic functions: • f(x) = x² • g(x) = x² - 4 • h(x) = x² + 3 • j(x) = -x² + 4

13. y y y x x x Match the Function with the Graph f(x) = x² + 4x + 4 g(x) = 0.2x² + 4 h(x) = -x² + 3x g(x) f(x) h(x)

14. Summary and Homework • Summary • Quadratic function: y = ax² + bx + c • Graph of a quadratic function is a parabola • The a coefficient determines the width and direction of the parabola • If b = 0, then the turning point is on the y-axis; if b ≠ 0, then the turning point won’t be on the y-axis • The c term is always the y-intercept of the parabola • Homework • page 416 – 420; problems 1-3, 7-11, 14