1. 1.2 day 2 Transformations of Functions
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2. 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.

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

4. 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.

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

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

7. 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.

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

9. 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.

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

11. 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.

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

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