Quadratic Functions and Models Lesson 3.1

Nonlinear Data When the points of the function are plotted, they do not lie in a straight line. This graph contains points from a quadratic function of the form Contrast to linear

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

Axis of Symmetry Parabolas are symmetric about a vertical axis For y = ax2 + bx + c the axis of symmetry is at Given y = 3x2 + 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) = 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)

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)

Other Quadratic Forms Standard form y = ax2 + bx + c Vertex form y = 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

Identifying Quadratic Functions How can you determine which of the following is quadratic or not? What determines whether a parabola opens down or up?

Modeling Quadratic Data Consider the following table … Is it linear or non linear? Graph the data on your calculator

Modeling Quadratic Data The calculator can do quadratic regression to find a modeling function While in the data matrix mode Press F5, then specify QuadReg Fill in remaining parameters as specified here.

Modeling Quadratic Data Note results Formula Graph

Assignment Lesson 3.1A Page 168 Exercises 1 – 47 odd Lesson 3.1B