Transformations of Quadratic Functions (9-3) Objective: Apply translations of quadratic functions. Apply vertical stretches and reflections to quadratic functions.

Translations A transformation changes the position or size of a figure. One transformation, a translation, moves a figure up, down, left or right. When a constant c is added to or subtracted from the parent function, the graph of the resulting function f(x) ± c is the graph of the parent function translated up or down. The parent function of the family of quadratics is f(x) = x2. All other quadratic functions have graphs that are transformations of the graph of f(x) = x2.

Vertical Translations The graph of f(x) = x2 + c is the graph of f(x) = x2 translated vertically. If c > 0, the graph of f(x) = x2 is translated c units up. If c < 0, the graph of f(x) = x2 is translated c units down. c > 0 c = 0 c < 0

Example 1 Describe how the graph of each function is related to the graph of f(x) = x2. h(x) = 10 + x2 Translated up 10 units. g(x) = x2 – 8 Translated down 8 units.

Check Your Progress Choose the best answer for the following. Describe how the graph of h(x) = x2 + 7 is related to the graph of f(x) = x2. h(x) is translated 7 units up from f(x). h(x) is translated 7 units down from f(x). h(x) is translated 7 units left from f(x). h(x) is translated 7 units right from f(x).

Check Your Progress Choose the best answer for the following. Describe how the graph of g(x) = x2 – 3 is related to the graph of f(x) = x2. g(x) is translated 3 units up from f(x). g(x) is translated 3 units down from f(x). g(x) is translated 3 units left from f(x). g(x) is translated 3 units right from f(x).

Dilations and Reflections Another type of transformation is a dilation. A dilation makes the graph narrower than the parent graph or wider than the parent graph. When the parent function f(x) = x2 is multiplied by a constant a, the graph of the resulting function f(x) = ax2 is either stretched or compressed vertically.

Dilations The graph of g(x) = ax2 is the graph of f(x) = x2 stretched or compressed vertically. If a > 1, the graph of f(x) = x2 is stretched vertically. If 0 < a < 1, the graph of f(x) = x2 is compressed vertically. a > 1 a = 1 0 < a < 1

Example 2 Describe how the graph of each function is related to the graph of f(x) = x2. d(x) = 1/3 x2 Compressed vertically m(x) = 2x2 + 1 Stretched vertically Translated up 1 unit

Check Your Progress Choose the best answer for the following. Describe how the graph of n(x) = 2x2 is related to the graph of f(x) = x2. n(x) is compressed vertically from f(x). n(x) is translated 2 units up from f(x). n(x) is stretched vertically from f(x). n(x) is stretched horizontally from f(x).

Check Your Progress Choose the best answer for the following. Describe how the graph of b(x) = ½ x2 – 4 is related to the graph of f(x) = x2. b(x) is stretched vertically and translated 4 units down from f(x). b(x) is compressed vertically and translated 4 units down from f(x). b(x) is stretched horizontally and translated 4 units up from f(x). b(x) is stretched horizontally and translated 4 units down from f(x).

Dilations and Reflections A reflection flips a figure across a line. When f(x) = x2 or the variable x is multiplied by -1, the graph is reflected across the x- or y-axis.

Reflections The graph of -f(x) is the reflection of the graph of f(x) = x2 across the x-axis. The graph of f(-x) is the reflection of the graph of f(x) = x2 across the y-axis. f(x) f(-x) -f(x)

Example 3 Describe how the graph of g(x) = -3x2 + 1 is related to the graph of f(x) = x2. Reflected over the x-axis. Stretched vertically. Translated up 1 unit.

Check Your Progress Choose the best answer for the following. Describe how the graph of g(x) = -5x2 – 4 is related to the graph of f(x) = x2. The graph of g(x) is reflected across the x-axis, compressed, and translated up 4 units. The graph of g(x) is reflected across the x-axis, compressed, and translated up 5 units. The graph of g(x) is reflected across the x-axis, stretched, and translated down 4 units. The graph of g(x) is reflected across the y-axis, and translated down 4 units

Transformations You can use what you know about the characteristics of graphs of quadratic equations to match an equation with a graph.

Example 4 Which is an equation for the function shown in the graph? y = 1/3 x2 – 2 y = 3x2 + 2 y = -1/3 x2 + 2 y = -3x2 – 2 Compressed Translated down 2 units

Check Your Progress Choose the best answer for the following. Which is an equation for the function shown in the graph? y = -2x2 – 3 y = 2x2 + 3 y = -2x2 + 3 y = 2x2 – 3 Reflected across the x-axis Stretched Translated up 3 units