Family of Quadratic Functions Lesson 5.5a

General Form Quadratic functions have the standard form y = ax 2 + bx + c  a, b, and c are constants  a ≠ 0 (why?) Quadratic functions graph as a parabola

Zeros of the Quadratic Zeros are where the function crosses the x- axis  Where y = 0 Consider possible numbers of zeros None (or two complex) One Two 

Axis of Symmetry Parabolas are symmetric about a vertical axis For y = ax 2 + bx + c the axis of symmetry is at Given y = 3x 2 + 8x  What is the axis of symmetry?

Vertex of the Parabola The vertex is the “point” of the parabola  The minimum value  Can also be a maximum What is the x-value of the vertex? How can we find the y-value?

Vertex of the Parabola Given f(x) = x 2 + 2x – 8 What is the x-value of the vertex? What is the y-value of the vertex? The vertex is at (-1, -9)

Vertex of the Parabola Given f(x) = x 2 + 2x – 8  Graph shows vertex at (-1, -9) Note calculator’s ability to find vertex (minimum or maximum)

Shifting and Stretching Start with f(x) = x 2 Determine the results of transformations  ___ f(x + a) = x 2 + 2ax + a 2  ___ f(x) + a = x 2 + a  ___ a * f(x) = ax 2  ___ f(a*x) = a 2 x 2 a) horizontal shift b) vertical stretch or squeeze c) horizontal stretch or squeeze d) vertical shift e) none of these

Other Quadratic Forms Standard form y = ax 2 + bx + c Vertex form y = a (x – h) 2 + k  Then (h,k) is the vertex Given f(x) = x 2 + 2x – 8  Change to vertex form  Hint, use completing the square Experiment with Quadratic Function Spreadsheet

Vertex Form Changing to vertex form Add something in to make a perfect square trinomial Subtract the same amount to keep it even. Now create a binomial squared This gives us the ordered pair (h,k)

Assignment Lesson 5.5a Page 231 Exercises 1 – 25 odd