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Graphs of Quadratic Equations

Module 3 Lesson 5:. Graphs of Quadratic Equations. a. h. k. Table of Contents Slides 3-9: Review Standard Form Slides 10-14: Review Vertex Form Slides 15-21: Review Transformations Slide 22: Find the Equation From a Graph

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Graphs of Quadratic Equations

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  1. Module 3 Lesson 5: Graphs of Quadratic Equations a h k

  2. Table of Contents • Slides 3-9: Review Standard Form • Slides 10-14: Review Vertex Form • Slides 15-21: Review Transformations • Slide 22: Find the Equation From a Graph • Slide 23: Find the Vertex and y-Intercept from an Equation • Audio/Video and Interactive Sites • Slide 4: Gizmos • Slide 6: Gizmos • Slide 12: Gizmos

  3. Explore Quadratic Equations In Standard Form

  4. For a more straightforward explanation on the affect of A, B, and C. Look at the following Gizmo. Gizmo: How A, B, and C affect the quadratic function. To explore the affect the variables A, B, and C have on the graph, play the Zap It game found at the following link. Zap It Game

  5. On your graphing calculator… Graph y= x2 Graph y = -x2 What do you see? What is A in the above two equations and how does it affect the graph?

  6. In this equation, A = -1 In this equation, A = 1 Gizmo: Standard Form, Vertex, and Intercepts Gizmo: Standard Form, Vertex, and Intercepts (B)

  7. f(x) = Ax2+ Bx + C If A = 0, the graph is not quadratic, it is linear. If A is not 0, B and/or C can be 0 and the function is still a quadratic function. If A < 0, The parabola opens down, like an umbrella. If A > 0, The parabola opens up, like an bowl. In this equation, A = -1 In this equation, A = 1

  8. When "a" is positive, the graph of y = ax2 + bx + c opens upward and the vertex is the lowest point on the curve. As the value of the coefficient "a" gets larger, |a| > 1, the parabola narrows. As the value of the coefficient “a” gets smaller, | a | is between 0 and 1, the parabola widens.

  9. For the graph of y = Ax2 + Bx + C If C is positive, the graph shifts up C units If C is negative, the graph shifts down C units Since C is negative, -5, the graph shifts down 5 units Since C is positive, 12, the graph shifts up 12 units. Note: This graph would open upside down since A is -2 Since C is negative, -8, the graph shifts down 8 units

  10. Vertex Form of the Quadratic Equation

  11. The Vertex Form of quadratic functions is often used, especially when we want to see how a graph behaves as values change. y=Ax2 + Bx + C (Standard Form) Can be factored y = a(x - h)2 + k (Vertex Form) Notice: If you are given the equation in Vertex Form, you can FOIL the (x-h) part, distribute a, and then combine like terms to get back to Standard Form.

  12. Gizmo: Vertex Form, Vertex, Intercepts Gizmo: Vertex Form, Vertex, Intercepts (B)

  13. Practice Assignment 1—State whether each Quadratic Function is in Vertex Form, Quadratic Form, or Neither. • Quadratic Form • b. Vertex Form • c. Neither

  14. Practice Assignment 1 Answers—State whether each Quadratic Function is in Vertex Form, Quadratic Form, or Neither. • Quadratic Form • b. Vertex Form • c. Neither

  15. Transformation affects of a, h, and k.

  16. When a is positive Graph opens up. As a increases, graph gets narrower As a decreases, graph gets wider When a and h are positive: a(x – (+h))2 = a(x – h)2 Graph moves right h units. When a is negative and h is positive: -a(x – (+h))2 = -a(x – h)2 Graph moves right h units. When k is positive Graph moves up regardless of what a and h are. When k is negative Graph moves down regardless of what a and h are. When a is positive and h is negative: a(x – (-h))2 = a(x + h)2 Graph moves left h units. When a is negative Graph opens down. As a increases (approaches 0), graph gets wider As a decreases (approaches negative infinity), graph gets narrower When a and h are negative: -a(x – (-h))2 = a(x + h)2 Graph moves left h units.

  17. When a is positive Graph opens up. As a increases, graph gets narrower As a decreases, graph gets wider

  18. When a is negative Graph opens down. As a increases (approaches 0), graph gets wider As a decreases (approaches negative infinity), graph gets narrower

  19. +k When k is positive Graph moves up regardless of what a and h are. - k When k is negative Graph moves down regardless of what a and h are.

  20. When a and h are positive: a(x – (+h))2 = a(x – h)2 Graph moves right h units. When a is negative and h is positive: -a(x – (+h))2 = -a(x – h)2 Graph moves right h units.

  21. When a is positive and h is negative: a(x – (-h))2 = a(x + h)2 Graph moves left h units. When a and h are negative: -a(x – (-h))2 = a(x + h)2 Graph moves left h units.

  22. Find the equation of the following graph. Start by writing down the generic vertex form equation. Note: a will be a negative since the graph is upside down. Identify the vertex: (1, 2) Identify the y-intercept: (0, -1) Fill in all the information you just found and solve for a. From the vertex: h = 1 and k = 2 From the y-intercept: x = 0 and y = -1 Simplify and Solve for a Therefore, the equation in vertex form is:

  23. Identify the vertex and y-intercept of the following equation. Identify the vertex: Identify the y-intercept: (0, -10.5) (h, k) (-3, -6)

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