1. Core Focus on Functions & Data
Lesson 1.5
Core Focus on Functions & Data
Function Reflections

2. Warm-Up
Describe the direction and amount of the translation that occurs to g(x). 1. g (x + 1) 2. g (x) g (x − 2) − 5 4. −2 + g (x + 4)
Left 1 unit
Up 7 units
Right 2 units, down 5 units
Left 4 units, down 2 units

3. Lesson 1.5
Function Reflections
Perform reflections on graphs and write equations to represent the reflections.

4. Vocabulary
Reflection A transformation in which a mirror image is produced by flipping a figure over a line.

5. Explore! Mirror Image
Willie graphed the letter W on a coordinate plane as shown at right.
Step 1 Is this graph a function? Why or why not?
Step 2 There are five key points on Willie’s graph that must be connected to form the W. List these points in a table of values in the order they should be connected. Start with the point shown below.
Step 3 Willie decided to see what happens if he keeps the x-values the same but changes all f(x) values (or y-values) to their opposites. Create a table of values for his transformation and then graph the points. How does the new graph compare to the original?

6. Explore! Mirror Image
Step 4 Next, Willie wants to see what would have happened if he changed the x-values in the original function to their opposites but left the f(x) values unchanged. Create a table of values for this transformation and then graph the points. How does this graph compare to the original?
Step 5 Summarize your findings by completing the sentences:
A graph is reflected over the x-axis when…
A graph is reflected over the y-axis when…

7. Reflecting Functions Reflection over the x-axis:
A reflection over the x-axis occurs when values of the entire function are changed to their opposites for each input.
Reflection over the y-axis:
A reflection over the y-axis occurs when the input is changed to its opposite for each original output.

8. Example 1
Use the graph of f(x) shown below. Create a new graph which has undergone each of the following transformations: a. f (−x)
Original
f(–x)
f(x)
When the x-values (input values) are changed to their opposites, a reflection over the y-axis occurs.

9. Example 1 Continued…
Use the graph of f(x) shown below. Create a new graph which has undergone each of the following transformations: b. −f (x)
f(x)
Original
–f(x)
When the output values (y-values or f (x) values) are changed to their opposites, a reflection over the x-axis occurs.

10. Good to Know!
You can write equations for the reflection of a graph over the x- or y-axis by inserting a negative sign into the equation.  For a reflection over the x-axis, a negative sign should be inserted in front of the entire equation. Original Equation Reflection Over the x-Axis If there are multiple terms in an equation, group them together with parentheses and then place the negative sign outside the parentheses and simplify.

11. Good to Know! Continued…
 For a reflection over the y-axis, a negative sign should be inserted in front of every term that includes an x-variable. If a term is being taken to a power, replace x with –x inside a set of parentheses. Original Equation Reflection Over the y-Axis

12. Example 2
Use the function f(x) = 4x2 − 5. a. Write a function, g(x), that is a reflection of f (x) over the y-axis.
A reflection over the y-axis means all x-values will change to their opposites.
f(x) = 4x2 − 5  g(x) = 4(−x)2 − 5
Notice x has an exponent so the negative sign is placed in parentheses with the x-value.

13. To simplify, distribute the negative to all terms.
Example 2 Continued…
Use the function f(x) = 4x2 − 5. b. Write a function, h(x), that is a reflection of f(x) over the x-axis.
A reflection over the x-axis means all y-values will change to their opposites.
f(x) = 4x2 − 5  h(x) = − (4x2 − 5) = −4x2 + 5
To simplify, distribute the negative to all terms.

14. Explain the difference between a reflection and a translation.
Communication Prompt
Explain the difference between a reflection and a translation.

15. Exit Problems
1. Below is a function. Graph this function after a reflection over the x-axis. 2. If the parent function above is f (x), what expression represents the reflection?
–f (x)