Lesson 9-3: Transformations of Quadratic Functions

Transformation A transformation changes the position or size of a figure 3 types of transformations: Translations Dilations Reflections

Vocabulary A dilation is a transformation that makes the graph narrower or wider than the parent graph. A reflection flips a figure over the x-axis or y-axis.

Dilations Raise your hand if you’ve ever been to the eye doctor. How many of you have had the eye doctor put some drops in your eyes to take a closer look at them? Okay, well the drops enlarge your pupils so that the doctor can look at them closely. Today, we’ll look at how we can stretch and compress parabolas.

Example 1: Describe how the graph of d(x) = x2 is related to the graph f(x) = x2. __ 1 3 Answer: Since 0 < < 1, the graph of f(x) = x2 is a vertical compression of the graph y = x2. __ 1 3

Example 2: Describe how the graph of m(x) = 2x2 + 1 is related to the graph f(x) = x2. Answer: Since 1 > 0 and 3 > 1, the graph of y = 2x2 + 1 is stretched vertically and then translated up 1 unit.

Example 3: Describe how the graph of n(x) = 2x2 is related to the graph of f(x) = x2. A. n(x) is compressed vertically from f(x). B. n(x) is translated 2 units up from f(x). C. n(x) is stretched vertically from f(x). D. n(x) is stretched horizontally from f(x).

Example 4: Describe how the graph of b(x) = x2 – 4 is related to the graph of f(x) = x2. __ 1 2 A. b(x) is stretched vertically and translated 4 units down from f(x). B. b(x) is compressed vertically and translated 4 units down from f(x). C. b(x) is stretched horizontally and translated 4 units up from f(x). D. b(x) is stretched horizontally and translated 4 units down from f(x).

Reflections

Three transformations are occurring: Example 1: How is the graph of g(x) = –3x2 + 1 related to the graph of f(x) = x2 ? Three transformations are occurring: First, the negative sign causes a reflection across the x-axis. Then a dilation occurs, where a = 3. Last, a translation occurs, where h = 1.

Answer:. g(x) = –3x2 + 1 is reflected across the Answer: g(x) = –3x2 + 1 is reflected across the x-axis, stretched by a factor of 3, and translated up 1 unit.

Example 2: Describe how the graph of g(x) = x2 – 7 is related to the graph of f(x) = x2. __ 1 5 Answer: (1/5) < 1, so the graph is vertically compressed and k = -7, so the graph is translated down 7 units

Example 2: Which is an equation for the function shown in the graph? A. y = –2x2 – 3 B. y = 2x2 + 3 C. y = –2x2 + 3 D. y = 2x2 – 3

Summary