### 2.3 Stretching, Shrinking, and Reflecting Graphs

```2.3 Stretching, Shrinking, and
Reflecting Graphs
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
Vertical Stretching
 Discussion
y
f(x) = x2
f(x) = 2x2
f(x) = 3x2
f(x) = 4x2
Notice: the x
intercept doesn’t
change.
x
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.
Vertical Shrinking
 Discussion
y
f(x) = x2
f(x) = ⅟2x2
f(x) = ⅟3x2
f(x) = ⅟4x2
Notice: The xintercept doesn’t
change, either.
x
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.
Vertical Stretching
and
Shrinking
y
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)
Horizontal Stretching and Shrinking
 Discussion
y
f(x) = |x|-1
f(x) = |⅟2x|-1
f(x) = |2x|-1
x
Notice: the yintercept doesn’t
change.
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).
Horizontal Stretching
and
Shrinking
y
0<c<1
y = f(cx)
c>1
y = f(cx)
c>1
y = f(cx)
0<c<1
y = f(cx)
x
Points on the y-axis
doesn’t change
Reflection
y
f(x) = x-3
g(x) = -x-3
g(x)
f(x)
h(x) = -x+3
x
g(x) = f(-x)
h(x) = -f(x)
h(x)
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
y
y = f(-x)
x
y = -f(x)
Combining Transformations of graphs
 Example 1
y = - ⅟2|x - 4| + 3
Step 1: Identify the basic function.
Step 2: shift right by 4 units.
Step 3: vertical shrink by ½.
Step 4: reflection across y-axis.
Step 5: shift up by 3 units.
f(x) = |x|
f(x) = |x-4|
f(x) = ½|x-4|
f(x) = -½|x-4|
f(x) = -½|x-4|+3
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!
```