Section 3.3
Graphing Techniques: Transformations
Horizontal and Vertical Shifts
 We investigated what the graph
f ( x)  x
 Let’s also graph g ( x)  x  2 and h( x)  x 1
x
f(x)
x
g(x)
x
h(x)
-2
2
-2
4
-2
3
-1
1
-1
3
-1
2
0
0
0
2
0
1
1
1
1
3
1
0
2
2
2
4
2
1
Horizontal and Vertical Shifts (cont.)
 We can notice that the graph of g(x) looks
very similar to f(x) but moved on the
coordinate plane.


How did it shift?
Up two spaces
 h(x) also looks similar to f(x) but shifted.


How did it shift?
Right one space
 f(x) is called the “Parent Function”
Vertical Shift
 Given the graph of some parent function f(x)…

To graph f(x) + c


Shift c units upward
To graph f(x) – c

Shift c units downward
Adding or subtracting a constant outside the
parent function corresponds to a vertical shift
that goes with the sign.
Examples of “outside the function”
x 2
x 4
3
Horizontal Shift
 Given the graph of some parent function f(x)…

To graph f(x + c)


Shift c units left
To graph f(x – c)

Shift c units right
Adding or subtracting a constant inside the
parent function corresponds to a horizontal
shift that goes opposite the sign.
Examples of “inside the function”
x  3
2
x 1
x2
Explain how this graph shifts
compared to its parent graph
g ( x)  x  2
Up 2
Right 3
h( x)  x  2
3
Left 2
f ( x)  x  3
j ( x)  x  4
Down 4
Given the parent function f(x) = x2, write
a new equation with the following shifts.
 Shift up 4 units
 Add 4 units outside the function
 f(x) + 4 = x2 + 4
 Shift right 1 unit
 Subtract 1 unit inside the function
 f(x – 1) = (x – 1)2
 Shift down 3 units, and left 2 units
 Subtract 3 units outside, and add 2 units
inside the function
 f(x+2) – 3 = (x + 2)2 – 3
Reflections about the Axes
 Let’s look at the graph of
 Now graph
f ( x)  x
again.
 f ( x)   x
x
f(x)
x
y
-2
2
-2
-2
-1
1
-1
-1
0
0
0
0
1
1
1
-1
2
2
2
-2
Reflection
 Note the graph of f ( x)  x
is reflected about the
x-axis, and the result is the graph of  f ( x)   x
 So with a given function f(x), to flip over the x –
axis, use –f(x)
 A negative symbol in front of the parent graph flips
the graph over the x-axis.
 This should make sense from Chapter 2. When
reflecting over the x-axis, we flip the sign of all y
values, which is exactly what we did in the
previous example.
Reflection
 Similarly, when reflecting over the y – axis,
we simply replace x with –x.

i.e. g(x) = f(-x)
 Graph f ( x)  x and g ( x)   x
 So to flip f(x) over the
y-axis, evaluate f(-x)
 A negative ON the x
flips the graph over the
y-axis.
Examples of Reflection
 Reflect over x-axis
 Reflect over y-axis
f ( x)  x3
When shifting a graph…
 Follow this order:



1.
2.
3.
Horizontal Shifts
Reflection over x or y axis
Vertical Shifts
Stretching and Compressing
f x   x 2
and hx   1 x 2
2
 Let’s look at the graph of
2
 Now graph g x   2x
x
f(x)
x
g(x)
x
g(x)
-2
4
-2
8
-2
2
-1
1
-1
2
-1
1
0
0
0
0
0
0
1
1
1
2
1
1
2
4
2
8
2
2
2
2
Vertical Stretch and Compress
 The graph of

Vertically stretching the graph of f(x)


c  f x is found by
If c > 1
Vertically compressing the graph of f(x)

If 0 < c < 1
Write the function whose graph is the graph of
f(x) = x3 with the following transformations.
 Vertically Stretched by a factor of 2

J(x) = 2x3
 Reflected about the y – axis

G(x) = (-x)3
 Vertically compressed by a factor of 3

H(x) = (1/3)x3
 Shifted left 2 units, reflected about x – axis

K(x) = -(x + 2)3
Use the given graph to sketch the
indicated functions.
 y = f(x + 2)
Use the given graph to sketch the
indicated functions.
 y = -f(x – 2)
Use the given graph to sketch the
indicated functions.
 y = 2f(–x)
Sketch the graphs of the following functions
using horizontal and vertical shifting.
 g(x) = x2 + 2
 The 2 is being added “outside” the function

Shifts up 2 units from parent function f(x) = x2
Sketch the graphs of the following functions
using horizontal and vertical shifting.
 h(x) = (x + 2)2
 The 2 is being added “inside” the function

Shifts 2 units left from parent function f(x) = x2
Sketch the graphs of the following functions
using horizontal and vertical shifting.
 g(x) = (x – 3)2 + 2

Shifts right 3 units and up 2 units from f(x) = x2
Sketch G(x) = -(x + 2)2
 Start with parent graph f(x) = x2
 Shift the graph 2 units left to obtain
f(x + 2) = (x + 2)2
 Reflect over the x – axis to obtain
–f(x+2) = -(x + 2)2
Graph
y   x3 2
Horizontal Stretch and Compress
 Similarly when multiplying by a constant c
“inside” the function
 The graph of f(cx) is found by:

Horizontally stretching the graph of f(x)


If 0 < c < 1
Horizontally compressing the graph of f(x)

c>1