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Chapter 9

Chapter 9. Quadratic Equations and Functions. 9.1 Graph y = ax 2 + c (b = 0). y. Vertex. x. Vertex. Quadratic Functions. parabola. The graph of a quadratic function is a:. A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex ( minimum ).

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Chapter 9

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  1. Chapter 9 Quadratic Equations and Functions

  2. 9.1 Graph y = ax2 + c (b = 0)

  3. y Vertex x Vertex Quadratic Functions parabola The graph of a quadratic function is a: A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex (minimum). If the parabola opens down, the vertex is the highest point (maximum). NOTE: if the parabola opens left or right it is not a function!

  4. y a > 0 x a < 0 The standard form of a quadratic function is: y = ax2 + bx + c The parabola will open up when the a value is positive. The parabola will open down when the a value is negative.

  5. Axis of Symmetry y x Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the Axis of symmetry. The Axis of symmetry ALWAYS passes through the vertex.

  6. Steps to Graphing Quadratic Functions 1) Find the Axis of symmetry using: 2) Find the vertex by using x to find y 3) Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.

  7. y x Graph y = x2 (parent function)

  8. y x Graph y = -3x2

  9. y x Graph y = ½x2 + 1

  10. Homework: 9.1 Practice

  11. 9.2 Graph y = ax2 + bx + c

  12. Axis of Symmetry y = ax2 + bx + c, When a quadratic function is in standard form the equation of the Axis of symmetry is Find the Axis of symmetry for y = 3x2 – 18x + 7 a = 3 b = -18 The Axis of symmetry is x = 3.

  13. Finding the Vertex The Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex. Vertex X-coordinate Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex.

  14. Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. STEP 1: Find the Axis of symmetry using: STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry.

  15. y x Graphing a Quadratic Function STEP 1: Find the Axis of symmetry STEP 2: Find the vertex STEP 3: Make table of values around vertex

  16. y x Example: Graph y = -x2 – 2x + 1

  17. Homework: p.580 #1, 2, 3 – 35odd

  18. 9.3 Solve Quadratic Equations by Graphing

  19. Solving Equations When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts. These values are also known as solutions, zeros, or roots.

  20. Solving a Quadratic The x-intercepts (when y = 0)of a quadratic function are the solutions to the related quadratic equation. The number of real solutions is at most two. Two solutions X= -2 or X = 2 One solution X = 3 No solutions

  21. Identifying Solutions Find the solutions of 2x - x2 = 0 The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2 The x-intercepts(or Zeros) of f(x)= 2x – x2 are the solutions to 2x - x2 = 0 X = 0 or X = 2

  22. y x Example: Solve the equation x2 + 5x + 6 = 0 by graphing

  23. Homework: p. 589 #1, 2, 3 – 39 odd

  24. 9.4 Use Square Roots to Solve Quadratic Equations I can solve a quadratic equation by finding square roots I can solve a problem about a falling object CC9-12.A.REI.4b

  25. When b = 0, the equation becomes y = ax2 + c You can use Square Roots Method to solve the equation (extracting roots)

  26. Steps to Solving Quadratics Using Square Roots Method Make sure b = 0 Get variable term on one side, constant on the other Get variable by itself Take square root of both sides

  27. Example: Solve 2x2 = 8

  28. Example: x2 + 12 = 7 Example: x2 + 12 = 25

  29. Example: A pinecone drops from a tree 150 feet up. How long will it take the pinecone to reach the ground.

  30. Homework: p. p.597 #1, 2, 3 – 49 odd, 56,

  31. 9.6 Solve Quadratic Equations by Using Quadratic Formula CC.9-12.A.REI.4b I can solve quadratic equations using the quadratic formula.

  32. Quadratic Formula:

  33. Example: Solve 3x2 + 5x = 8

  34. Example: Solve 2x2 – 7 = x

  35. Example: x2 – 8x + 16 = 0

  36. Methods for Solving Quadratic Equations Some methods are better than others depending on situation Factoring – good when quadratic eqn can be factored easily Graphing – use when approximate solutions are fine Square roots method – use when b = 0 (no bx term) Quadratic formula – can be used at any time!!!

  37. Example: tell what method you would use to solve the quadratic equation. Explain. 3x2 – 27 = 0 x2 + 5x + 6 = 0

  38. Homework: p.616 #1, 2, 3 – 43 odd, 47, 48

  39. 9.6 extended – The Discriminant

  40. 9.5 Solve Quadratic Equations by Completing the Square

  41. 9.7 Solve Systems of Quadratic Equations

  42. 9.8 Compare Linear, Exponential, and Quadratic Functions

  43. 9.9 Model Relationships

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