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:

1. Functions – Translations
2. Functions – Symmetries

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?