1.2 day 2 Transformations of Functions Page 12

The rules for shifting, stretching, shrinking, and reflecting the graph of a function make it easier to sketch functions by hand. Since we will be frequently using graphs in our study of calculus, we will do a quick review of those rules. If we know how to graph the “parent” graph of a function, then we can modify that graph to get the one we want.

Example: Adding a positive number at the end moves the graph up.

Example: Adding a constant to x inside the parentheses moves the graph to the left. The horizontal changes happen in the opposite direction to what you might expect.

Example: Placing a coefficient in front of the function causes a vertical stretch. In this case, the graph goes up twice as fast.

Example: If the coefficient is negative, then the graph is reflected about the x-axis.

Example: Placing a coefficient inside the function in front of the x causes a horizontal shrink. In this case, the graph expands horizontally half as fast. The horizontal changes happen in the opposite direction to what you might expect.

Example: Clearing the parentheses: In this case, a horizontal shrink is the same as a vertical stretch, but this is not always true.

Example: If the coefficient inside the function in front of the x is negative, you get a reflection about the y axis. In this case, since we started with an even function, we cannot see the reflection. Let’s look at an odd function.

Example: Placing a negative coefficient inside the function in front of the x causes a reflection about the y-axis.

To summarize the rules for transformations of graphs: Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. is a stretch. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. is a shrink. The horizontal changes happen in the opposite direction to what you might expect.

p Now let’s look at a more complicated example: vertical right three flip right three moves down half as fast up four p

Homework 1.2b 1.2 p19 6,12,22,25,28,31,44,49