Chapter 2 Quadratic Functions. How do we build quadratic functions? Take two linear functions and multiply them together It’s called multiplying binomials.

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Presentation transcript:

Chapter 2 Quadratic Functions

How do we build quadratic functions? Take two linear functions and multiply them together It’s called multiplying binomials For example, (x + 2) and (x – 3)

(3x + 2)(x + 3)(x – 1)(2x – 2) (2x + 3)(2x-3)(3x – 3)(x – 1)

(5x – 2)(5x – 8)(5x + 6)(8x – 4) (4x + 4)(5x – 1)(x – 2)(x – 3)

Characteristics of a Quadratic Function x-intercepts: where the parabola crosses the x- axis (can have 1, 2, or none) y-intercepts: Where the parabola crosses the y- axis (every function has one) Vertex: Tip of the parabola, either maximum or minimum Axis of symmetry: The line that cuts the parabola in half Concavity: Whether the graph opens up or down

Sorting Activity Match each function with its corresponding graph You may use a graphing calculator to find the matches Notice that they’re all quadratic functions, just in different forms

3 Forms of Quadratic Equations Standard Form: f(x) = ax 2 + bx + c Factored Form: f(x) = a(x – r 1 )(x – r 2 ) Vertex Form: f(x) = a(x – h) 2 + k Each form serves a different purpose. What do you think are the key characteristics you can determine from each form?

Warm Up

Converting Between Quadratic Function Forms From Factored to Standard Multiply the binomials Multiply by the coefficient (a) From Vertex to Standard Separate the square Multiply the binomials Multiply by the coefficient (a) Add c

Reference Points Set of key points that help identify the basic form of any function For a basic quadratic function, they’re P(0,0); Q(1,1); R(2,4)

How to Translate A Quadratic Function Find out the new vertex and plot Adjust the other two reference points the same amount that you adjusted the vertex For example, if your new vertex is at (2,8), that means you moved to the right 2 and up 8. Take your reference points of (1,1) and (2,4) and do the same to get (3,9) and (4,12).

Warm Up

Writing a Quadratic Equation Writing equations given roots and a point: Plug the given roots into factored form for r 1 and r 2 and the point for x and y. Solve for a. Put a, r 1 and r 2 back into factored form. Writing equations given the vertex and a point: Plug the vertex into vertex form for h and k and the point for x and y.