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Quad fcn

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Angie Starrett
Angie Starrett

This document discusses graphing quadratic functions. It covers the general form and vertex form of quadratic functions, as well as how to determine if a parabola opens up or down. It provides steps for graphing quadratic functions in both general and vertex form, including finding the vertex, x-intercepts, and y-intercept. An example of graphing a quadratic that models the path of a punted football is included. Key points about graphing quadratics are summarized.

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Quadratic Functions Graphing and Modeling
Quadratic Functions as Projectiles f (x)= ax2 +bx+c where a, b and c are real and a ¹ 0.
Characteristics of Graphs
Relating Solutions of the Quadratic Equation with x-intercepts
Relating Graphs with the Quadratic Formula
The Basic Quadratic and the Transformed Quadratic
Graphing Quadratic Function in Vertex Form – The Steps Determine whether the parabola opens UP or DOWN Determine the vertex (h,k) Find any x-intercepts by solving f(x)=0 Find the y-intercept by computing f(0) Plot the intercepts and vertex f (x) = a(x -h)2 +k
Graphing Quadratic Function in Vertex Form – An Example f (x) = -2(x -3)2 +8 a = -2 h = 3 k = 8 Since a is negative we know the parabola opens DOWN Since the vertex has the form (h, k), our vertex will be (3, 8)
Graphing Quadratic Function in Vertex Form – An Example Solving for the x- and y-intercepts
Graphing Quadratic Function in Vertex Form – An Example
Graphing Quadratic Function in General Form – The Steps Determine whether the parabola opens UP or DOWN Determine the vertex Find any x-intercepts by solving f(x)=0 Find the y-intercept by computing f(0) Plot the intercepts and vertex f (x) = ax2 +bx+c - b 2a , f - b 2a æ è ç ö ø ÷ æ è ç ö ø ÷
Graphing Quadratic Function in General Form – An Example f (x) = x2 +4x+1 a =1 b = 4 Since a is positive we know the parabola opens UP x-coordinate of the vertex: y-coordinate of the vertex: The vertex: -2,-3( )
Graphing Quadratic Function in General Form – An Example Solving for the x- and y-intercepts
Graphing Quadratic Function in General Form – An Example
The Parabolic Path of a Punted Football When a football is kicked, the height of the punted football, f(x), in feet, can be modeled by f (x)= -0.01x2 +1.18x+2 where x is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot. a. What is the maximum height of the punt? x = - b 2a = - 1.18 2(-0.01) = -(-59) = 59 feet
The Parabolic Path of a Punted Football When a football is kicked, the height of the punted football, f(x), in feet, can be modeled by f (x)= -0.01x2 +1.18x+2 where x is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot. a. What is the maximum height of the punt? f (59)= -0.01(59)2 +1.18(59)+2 The maximum height of the punt occurs 59 feet from the kicker’s point of impact. The actual maximum height of the punt is =36.81 feet
The Parabolic Path of a Punted Football Continued f (x)= -0.01x2 +1.18x+2 b. How far must the nearest defensive player, who is 6 feet from the kicker’s point of impact, reach to block the punt? This means we need to find the height of the ball 6 feet from the kicker. In other words, “plug in” 6 for x. f (6)= -0.01(6)2 +1.18(6)+2 =8.72 feet The defensive player must reach 8.72 feet above the ground to block the punt.
Key Points to Know for Graphing Quadratic Functions: General form versus Vertex form Understanding the shape of a quadratic function When a parabola opens up or down Using either form to graph a parabola Able to solve for a maximum or minimum Able to solve for x- and y-intercepts
I Challenge You… Write a quadratic function in standard form that models the area of the shaded region. x +9 x+5 x +1 x+3 x +3 x -1 x x

More Related Content

Quad fcn

  • 2. Quadratic Functions as Projectiles f (x)= ax2 +bx+c where a, b and c are real and a ¹ 0.
  • 4. Relating Solutions of the Quadratic Equation with x-intercepts
  • 5. Relating Graphs with the Quadratic Formula
  • 6. The Basic Quadratic and the Transformed Quadratic
  • 7. Graphing Quadratic Function in Vertex Form – The Steps Determine whether the parabola opens UP or DOWN Determine the vertex (h,k) Find any x-intercepts by solving f(x)=0 Find the y-intercept by computing f(0) Plot the intercepts and vertex f (x) = a(x -h)2 +k
  • 8. Graphing Quadratic Function in Vertex Form – An Example f (x) = -2(x -3)2 +8 a = -2 h = 3 k = 8 Since a is negative we know the parabola opens DOWN Since the vertex has the form (h, k), our vertex will be (3, 8)
  • 9. Graphing Quadratic Function in Vertex Form – An Example Solving for the x- and y-intercepts
  • 10. Graphing Quadratic Function in Vertex Form – An Example
  • 11. Graphing Quadratic Function in General Form – The Steps Determine whether the parabola opens UP or DOWN Determine the vertex Find any x-intercepts by solving f(x)=0 Find the y-intercept by computing f(0) Plot the intercepts and vertex f (x) = ax2 +bx+c - b 2a , f - b 2a æ è ç ö ø ÷ æ è ç ö ø ÷
  • 12. Graphing Quadratic Function in General Form – An Example f (x) = x2 +4x+1 a =1 b = 4 Since a is positive we know the parabola opens UP x-coordinate of the vertex: y-coordinate of the vertex: The vertex: -2,-3( )
  • 13. Graphing Quadratic Function in General Form – An Example Solving for the x- and y-intercepts
  • 14. Graphing Quadratic Function in General Form – An Example
  • 15. The Parabolic Path of a Punted Football When a football is kicked, the height of the punted football, f(x), in feet, can be modeled by f (x)= -0.01x2 +1.18x+2 where x is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot. a. What is the maximum height of the punt? x = - b 2a = - 1.18 2(-0.01) = -(-59) = 59 feet
  • 16. The Parabolic Path of a Punted Football When a football is kicked, the height of the punted football, f(x), in feet, can be modeled by f (x)= -0.01x2 +1.18x+2 where x is the ball’s horizontal distance, in feet, from the point of impact with the kicker’s foot. a. What is the maximum height of the punt? f (59)= -0.01(59)2 +1.18(59)+2 The maximum height of the punt occurs 59 feet from the kicker’s point of impact. The actual maximum height of the punt is =36.81 feet
  • 17. The Parabolic Path of a Punted Football Continued f (x)= -0.01x2 +1.18x+2 b. How far must the nearest defensive player, who is 6 feet from the kicker’s point of impact, reach to block the punt? This means we need to find the height of the ball 6 feet from the kicker. In other words, “plug in” 6 for x. f (6)= -0.01(6)2 +1.18(6)+2 =8.72 feet The defensive player must reach 8.72 feet above the ground to block the punt.
  • 18. Key Points to Know for Graphing Quadratic Functions: General form versus Vertex form Understanding the shape of a quadratic function When a parabola opens up or down Using either form to graph a parabola Able to solve for a maximum or minimum Able to solve for x- and y-intercepts
  • 19. I Challenge You… Write a quadratic function in standard form that models the area of the shaded region. x +9 x+5 x +1 x+3 x +3 x -1 x x

Editor's Notes

  1. Sports objects often follow these paths. Polynomial whose largest exponent is 2. SLO – use graphs of quadratics to gain geometric understanding of the algebra that appears in football, baseball, basketball, etc. Refer to general form
  2. Symmetry! Vertex. Opens up or down depending on sign of a
  3. Vertex form with tranformations
  4. Will the vertex be a max or a min?
  5. (5,0), (1,0), (0,-10)
  6. What’s different? Vertex & solving for x-intercepts.