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5-Minute Check on Activity 4-1 Click the mouse button or press the Space Bar to display the answers. 1.Which direction does y = 3x 2 open? 2.Which function’s.

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Presentation on theme: "5-Minute Check on Activity 4-1 Click the mouse button or press the Space Bar to display the answers. 1.Which direction does y = 3x 2 open? 2.Which function’s."— Presentation transcript:

1 5-Minute Check on Activity 4-1 Click the mouse button or press the Space Bar to display the answers. 1.Which direction does y = 3x 2 open? 2.Which function’s graph is narrower, f(x) =4x 2 or g(x) = ½x 2 ? Solve the following equations for x 3.8 = 2x 2 4.27 = 3x 2 5.y = 15 and y = 3x 2 up f(x) 4 = x 2  4 = x ± 2 = x 9 = x 2  9 = x ± 3 = x 15 = 3x 2 5 = x 2 ±  5 = x

2 Baseball and the Sears Tower Activity 4 - 2

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 the first second? How far does it fall during the 2 nd second? What is the average rate of change of H with respect to t in the first second? During the 2 nd second? Time, t (sec)Ht, H = -16t² + 1450 0 1 2 3 4 5 6 7 8 9 10 1450 1434 1386 1306 1194 1050 874 666 426 154 -150 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 domain of the height function? What is the practical range of the height function? Now graph the function using the table to the right Time, t (sec)Ht, H = -16t² + 1450 01450 11434 21386 31306 41194 51050 6874 7666 8426 9154 10-150 About 9.5 seconds 0 ≤ t ≤ 9.5 seconds 0 ≤ H ≤ 1450 feet

8 Activity Continued Is the shape of the curve the path of the ball? Time, t (sec)Ht, H = -16t² + 1450 01450 11434 21386 31306 41194 51050 6874 7666 8426 9154 10-150 H t 800 600 400 200 1400 1200 1000 No, the ball falls straight down

9 Quadratic Function Standard form: y = ax² + bx + c Quadratic term: ax² –Determines Direction a > 0 then parabola opens up a < 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 The Effects of a in y = ax² + bx + c Graph the following quadratic functions: a)f(x) = x² b)g(x) = ½x² c)h(x) = 2x² d)j(x) = -2x² y x

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

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

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


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