Before: March 19, 2018 For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 𝑦 = 𝑥 2 + 3 𝑦 = 2 𝑥 2 𝑦 = −0.5 𝑥 2 − 4

DURING: Transforming quadratic functions Learning Target: I can graph and transform quadratic functions.

Transforming Quadratic Functions The quadratic parent function is 𝑓(𝑥) = 𝑥 2 . The graph of all other quadratic functions are transformations of the graph of 𝑓(𝑥) = 𝑥 2 . For the parent function f(x) = 𝑥 2 : The axis of symmetry is x = 0, or the y-axis. The vertex is (0, 0) The function has only one zero, 0.

Transforming Quadratic Functions Previously, we learned when a > 0 (positive), the graph of a quadratic function opens upward. When a < 0 (negative), the graph of a quadratic function opens downward. This change in direction is a transformation: a reflection over the vertex.

Transforming Quadratic Functions The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola.

Transforming Quadratic Functions If |a| > 1, the transformation is a vertical stretch If |a| < 1, the transformation is a vertical compression

Transforming Quadratic Functions I Do: Are the graphs of the functions narrow (stretched) or wide (compressed)? 𝑦 =3 𝑥 2 𝑦 = 0.5 𝑥 2

Transforming Quadratic Functions We Do: Order the functions from narrowest graph to widest. 𝑦 = − 𝑥 2 𝑦 = 2 3 𝑥 2

Transforming Quadratic Functions You Do: Order the functions from narrowest graph to widest. 𝑦 = 1 2 𝑥 2 𝑦 = −2 𝑥 2

Transforming Quadratic Functions The value of c makes these graphs look different. The value of c in a quadratic function determines not only the value of the y-intercept but also a vertical translation of the graph of 𝑦 = 𝑎𝑥 2 up or down the y-axis.

Transforming Quadratic Functions This can also be notated as 𝑦 = 𝑥 2 + 𝑘. When comparing graphs, it is helpful to draw them on the same coordinate plane. Helpful Hint

Transforming Quadratic Functions If c > 0, the graph is translated up If c < 0, the graph is translated down

Transforming Quadratic Functions I Do: Compare the graph of the function with the graph of the parent function 𝑦 = 𝑥 2 . 1. 𝑦 = − 1 4 𝑥 2 +3

Transforming Quadratic Functions We Do: Compare the graph of the function with the graph of the parent function 𝑦 = 𝑥 2 . 1. 𝑦 =3 𝑥 2

Transforming Quadratic Functions You Do: Compare the graph of the function with the graph of the parent function 𝑦 = 𝑥 2 . 1. 𝑦 = − 𝑥 2 − 4