Parabolas and Shot Put: Analyzing Graphs and Equations

1. 5-Minute Check on Activity 4-2 In the general equation, y = ax2 + bx + c, identify what a, b, and c tell us about the graph. In the following equations, identify the direction it opens and what the y-intercept is for each y = -2x2 + 6x - 2 y = 2x2 + 12x + 4 y = 3x2 + 6x – 9 What is the domain of problem 2? a : if a > 0 then it opens up; otherwise it opens down b: helps determine the vertex and line of symmetry c: is the y-intercept of the parabola a < 0, so it opens down; y-intercept = -2 a > 0, so it opens up; y-intercept = 4 a > 0, so it opens up; y-intercept = -9 Domain: {x | x Real #s} Click the mouse button or press the Space Bar to display the answers.

2. Activity 4 - 3 The Shot Put

3. Objectives • Determine the vertex or turning point of a parabola • Identify the vertex as a maximum or a minimum • Determine the axis of symmetry of a parabola • Identify the domain and range • Determine the y-intercept of a parabola • Determine the x-intercept(s) of a parabola using technology • Interpret the practical meaning of the vertex and intercepts in a given problem • Identify the vertex from the standard form y = a(x – h)² + k of the equation of a parabola

4. Vocabulary • None new

5. Activity Parabolas are good models for a variety of situations that you encounter in everyday life. Examples include the path of a golf ball after it is struck, the arch (cable system) of a bridge, the path of a baseball thrown from the outfield to home plate, the stream of water from a drinking fountain, and the path of a cliff diver. Consider the 2000 men’s Olympic shot put event, which was won by Finland’s ArsiHarju with a throw of 69 feet 10¼ inches. The path of his winning throw can be approximately modeled by the quadratic function defined by Y = -0.015545x² + x + 6 Where x is the horizontal distance in feet from the point of the throw and y is the vertical height in feet of the shot above the ground.

6. Activity Y = -0.015545x² + x + 6 • Which way will the graph of the parabola open? • What is the y-intercept of the graph of the parabola? • What is the practical meaning of this value? • What is the practical domain of the graph? • What is the practical range of the graph? Since a (-0.015545)< 0, it opens down c  y-intercept so c = 6 The shot is 6 feet above the ground when its released 0 ≤ x ≤ 69’ 10 ¼ “ 0 ≤ y ≤ 22.03’

7. Vertex of a Parabola • A Parabola, y = ax2 + bx + c has a vertex at the coordinates:where a is the coefficient of the x2 term, b is the coefficient of the x-term and c is the y-intercept • If parabola opens: up, then vertex is a minimum down, then vertex is a maximum b b2 – 4ac - ------- , - ------------- 2a 4a

8. Example 1 Determine the vertex of y = -3x2 + 12x + 5 By Formula: By Calculator: b b2 – 4ac - ------- , - ------------- 2a 4a 12 122 – 4(-3)5 - ------- , - ------------------ 2(-3) 4(-3) (2, 17) Graph function (Y1=); 2nd Trace, select minimum; Move + so Left Bound on left side and enter Move + so Right bound on right side and enter Move + toward vertex and enter

9. Axis of Symmetry of a Parabola • A parabola, y = ax2 + bx + c has an axis of symmetry defined by the vertical line:where a is the coefficient of the x2 term, b is the coefficient of the x-term and c is the y-intercept • The axis of symmetry • A vertical line that passes through the x-value of the vertex coordinates • Divides the parabola into right and left halves -b x = ------- 2a

10. Example 2 Determine the axis of symmetry of y = -3x2 + 12x + 5 By Formula: x = 2 b 12 x = - ------- = - ------- 2a 2(-3)

11. X-intercepts of a Parabola • A parabola, y = ax2 + bx + c has x-intercepts if it crosses the x-axis (y = 0). A special equation will give us these if they exist. • Graphical Conditions (nonexistence): • If a parabola opens up and the vertex is above the x-axis, then there are no x-intercepts • If a parabola opens down and the vertex is below the x-axis, then there are no x-intercepts • X-intercepts are also known as the zeros of a function

12. Example 3 Determine if y = -3x2 + 12x + 5 has any x-intercepts? Determine if y = x2 + 12x + 6 has any x-intercepts? By its graph it crosses the x-axis between -1 and 0 and again between 4 and 5 By its graph it opens upward and it’s vertex is above the x-axis  no x-intercepts

13. Standard Form of a Parabola • A parabola, y = ax2 + bx + c is in standard form when the vertex can be read from the equation. • Parabola Standard Form: y = f(x) = a (x – h)2 + kwhere (h, k) are the coordinates of the vertex

14. Example 4 vertex (0, 0) Sketch y = 3x2 Sketch y = 3(x – 2)2  horizontal shift to the right Sketch y = 3(x -2)2 + 5  vertical shift up Where is the vertex for each? vertex (2, 0) vertex (2, 5) Remember: Horizontal shifts are inside and vertical shifts are outside the function

15. Domain and Range of a Parabola • Domain: all permissible x-values • Range: all possible y-values • A parabola, y = ax2 + bx + c has the following domain and rangeDomain: x = all real numbersRange: parabola opens up y ≥ y-value of vertexparabola opens down y ≤ y-value of vertex

16. Example 5 Determine the domain and range ofy = -3x2 + 12x + 5 Determine the domain and range of y = x2 + 12x + 6 Its vertex is (2, 17) and it opens downward Domain: x = all real numbers Range: y ≤ 17 Its vertex is (-3, 3) and it opens upward Domain: x = all real numbers Range: y ≥ 3

17. Summary and Homework • Parabola Summary So Far • Quadratic Form: y = f(x) = ax2 + bx + c • Standard Form: y = f(x) = a(x – h)2 + k • Vertex: (-b / [2a], -[b2 – 4ac] / [4a]) • Axis of Symmetry: x = -b/(2a) (Vertical Line) • Domain: x = all real numbers • Range: y ≥ min or y ≤ max • a determines if it opens up (a> 0) or down (a < 0) • Homework • pg 430 – 434; problems 1 – 4, 9, 10