Family of Quadratic Functions

1. Family of Quadratic Functions Lesson 5.5a

2. General Form • Quadratic functions have the standard form y = ax2 + bx + c • a, b, and c are constants • a ≠ 0 (why?) • Quadratic functions graph as a parabola

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

4. Axis of Symmetry • Parabolas are symmetric about a vertical axis • For y = ax2 + bx + c the axisof symmetry is at • Given y = 3x2 + 8x • What is the axis of symmetry?

5. Vertex of the Parabola • The vertex is the “point” of theparabola • The minimum value • Can also be a maximum • What is the x-value of thevertex? • How can we find the y-value?

6. Vertex of the Parabola • Given f(x) = x2 + 2x – 8 • What is the x-value of the vertex? • What is the y-value of the vertex? • The vertex is at (-1, -9)

7. Vertex of the Parabola • Given f(x) = x2 + 2x – 8 • Graph shows vertex at (-1, -9) • Note calculator’s ability to find vertex (minimum or maximum)

8. Shifting and Stretching • Start with f(x) = x2 • Determine the results of transformations • ___ f(x + a) = x2 + 2ax + a2 • ___ f(x) + a = x2 + a • ___ a * f(x) = ax2 • ___ f(a*x) = a2x2

9. Other Quadratic Forms • Standard formy = ax2 + bx + c • Vertex formy = a (x – h)2 + k • Then (h,k) is the vertex • Given f(x) = x2 + 2x – 8 • Change to vertex form • Hint, use completing the square Experiment with Geogebra Quadratic Function

10. 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) Vertex Form • Changing to vertex form

11. Assignment • Lesson 5.5a • Page 231 • Exercises 1 – 25 odd