1 / 17

Parabolas and Shot Put: Analyzing Graphs and Equations

Learn how to identify the direction, y-intercept, vertex, axis of symmetry, domain, range, and x-intercepts of a parabola. Apply this knowledge to analyze the graph of the shot put event in the Olympics.

jcabrera
Download Presentation

Parabolas and Shot Put: Analyzing Graphs and Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related