Table of Contents

Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 5: Basic functions and transformations

**Topic**. This lesson covers Session 5: Basic functions and transformations

**Learning Outcomes.**

- Translate between geometric transformations (shifting, stretching, flipping) in either direction (vertically, horizontally) and the corresponding algebraic transformations of a function
- Identify even and odd symmetries.

**WeBWorK**. There are two WeBWorK assignments on today’s material:

- Functions – Translations
- Functions – Symmetries

**Additional Video Resources.**

**Question of the Day**: When you flip a graph over, what happens to the formula?

## Transformations of graphs

**Shifting a graph up or down**: Add or subtract a number to the *output* , so .

- Consider the graph of a function . Then the graph of is that of shifted up or down by . If is positive, the graph is shifted up, if is negative, the graph is shifted down.

**Shifting a graph left or right**: Add or subtract a number to the *input* , so

- Consider the graph of a function . Then the graph of is that of shifted left or right by . Careful: If is positive, the graph is shifted
*left*, if is negative, the graph is shifted*right*.

**Stretching or compressing a graph vertically**: Multiply the *output* by a positive number , so

- Consider the graph of a function and let . Then the graph of is that of stretched away or compressed towards the -axis by a factor . If , the graph is stretched away from the -axis, if then the graph is compressed towards the -axis.

**Stretching or compressing a graph horizontally**: Multiply the *input* by a positive number , so

- Consider the graph of a function and let . Then the graph of is that of stretched away or compressed towards the -axis by a factor . If , the graph is stretched away from the -axis, if then the graph is compressed towards the -axis.

**Reflect a graph horizontally or vertically: **

- To reflect vertically (across the -axis), multiply the
*output*of the function by , so . - To reflect horizontally (across the -axis), multiply the
*input*of the function by , so .

*VIDEO: Transformations of functions*

*Video by Irania Vazquez*

## Symmetries: Odd and Even functions

**Definition**. A function is called **even** if for all .

Similarly, a function is called **odd** if for all

**Observation**.

- An even function is symmetric with respect to the -axis (if you reflect the graph horizontally across the -axis, you end up with the same graph – the left side is a mirror image of the right side).
- An odd function is symmetric with respect to the origin (if you rotate the graph about the origin, you end up with the same graph).

*VIDEO: Odd and Even Functions*

*Video by Irania Vazquez*

#### Exit Question

Is the function shown below odd or even?

#### Answer

The function is **even**, since the right side looks like a mirror image of the left side.

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.