1. 2.3 Stretching, Shrinking, and Reflecting Graphs

2. Quiz What’s the transformation of y = cf(x) if c>1 from y = f(x)?
A vertical shrinking
B vertical stretching
C horizontal shrinking
D horizontal stretching

3. Notice: the x intercept doesn’t change.
Vertical Stretching
Discussion
f(x) = x2 y
f(x) = 2x2
f(x) = 3x2
f(x) = 4x2
Notice: the x intercept doesn’t change. x

4. Vertical Stretching
If a point (x,y) lies on the graph of y = f(x), then the point (x, cy) lies on the graph of y = cf(x).
If c>1, then the graph of y = cf(x) is a vertical stretching of the graph of y = f(x) by applying a factor of c.

5. Notice: The x-intercept doesn’t change, either.
Vertical Shrinking y
Discussion
f(x) = x2
f(x) = ⅟2x2
f(x) = ⅟3x2
f(x) = ⅟4x2 x
Notice: The x-intercept doesn’t change, either.

6. Vertical Shrinking
If a point (x,y) lies on the graph of y = f(x), then the point (x,cy) lies on the graph of y = cf(x).
If 0 < c < 1, then the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x) by applying a factor of c.

7. Vertical Stretching and Shrinkingy
c>1 y = cf(x)
0<c<1
y = cf(x) x
0<c<1
y = cf(x)
Points on the x-axis doesn’t change
c>1 y = cf(x)

8. Horizontal Stretching and Shrinkingy
Discussion
f(x) = |x|-1
f(x) = |⅟2x|-1
f(x) = |2x|-1 x
Notice: the y-intercept doesn’t change.

9. Horizontal Stretching and Shrinking
If a point (x,y) lies on the graph of y = f(x), then the point (x/c, y) lies on the graph of y = f(cx)
If 0 < c < 1, then the graph of y = f(cx) is a horizontal stretching of the graph of y = f(x).
If c > 1, then the graph of y = f(cx) is a horizontal shrinking of the graph of y = f(x).

10. Horizontal Stretching and Shrinkingy
c>1
y = f(cx)
c>1
y = f(cx)
0<c<1
y = f(cx)
0<c<1
y = f(cx) x
Points on the y-axis doesn’t change

11. Reflection f(x) = x-3 g(x) = -x-3 h(x) = -x+3 g(x) = f(-x)y
f(x) = x-3
g(x) = -x-3
h(x) = -x+3
g(x)
f(x) x
g(x) = f(-x)
h(x) = -f(x)
h(x)

12. Reflection
The graph of y = -f(x) is a reflection of the graph of f across the x-axis.
The graph of y = f(-x) is a reflection of the graph of f across the y-axis x y
y = f(-x)
y = -f(x)

13. Combining Transformations of graphs
Example 1
y = - ⅟2|x - 4| + 3
Step 1: Identify the basic function f(x) = |x|
Step 2: shift right by 4 units f(x) = |x-4|
Step 3: vertical shrink by ½ f(x) = ½|x-4|
Step 4: reflection across y-axis f(x) = -½|x-4|
Step 5: shift up by 3 units f(x) = -½|x-4|+3

14. Homework
PG. 111: 24-36(M3), 38, 41, 48, 53, 58, 63, 64, 79, 90
KEY: 18, 27, 49, 51
No Reading, next class is review, so no quiz! But bring your student handbook!