Section 3.3

Graphing Techniques: Transformations

Horizontal and Vertical Shifts

We investigated what the graph Let’s also graph and

x g(x)

-2 4

-1 3

0 2

1 3

2 4

xxf )(

2)( xxg 1)( xxh

x f(x)

-2 2

-1 1

0 0

1 1

2 2

x h(x)

-2 3

-1 2

0 1

1 0

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”

2x 43 x

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”

23x 1x 2x

Explain how this graph shifts compared to its parent graph

2)( xxg

Up 2

3)( xxf

Right 3

32)( xxh 4)( xxj

Left 2 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 again. Now graph

xxf )(xxf )(

x f(x)

-2 2

-1 1

0 0

1 1

2 2

x y

-2 -2

-1 -1

0 0

1 -1

2 -2

Reflection

Note the graph of is reflected about the x-axis, and the result is the graph of

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.

xxf )(

xxf )(

Reflection

Similarly, when reflecting over the y – axis, we simply replace x with –x. i.e. g(x) = f(-x)

Graph and So to flip f(x) over the

y-axis, evaluate f(-x) A negative ON the x

flips the graph over the y-axis.

xxf )( xxg )(

Examples of Reflection

Reflect over x-axis

Reflect over y-axis

3)( xxf

When shifting a graph…

Follow this order: 1. Horizontal Shifts 2. Reflection over x or y axis 3. Vertical Shifts

Stretching and Compressing

Let’s look at the graph of Now graph and

x f(x)

-2 4

-1 1

0 0

1 1

2 4

x g(x)

-2 8

-1 2

0 0

1 2

2 8

x g(x)

-2 2

-1

0 0

1

2 2

21

21

2xxf

22xxg 2

2

1xxh

Vertical Stretch and Compress

The graph of is found by Vertically stretching the graph of f(x)

If c > 1 Vertically compressing the graph of f(x)

If 0 < c < 1

xfc

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

g(x) = (x – 3)2 + 2 Shifts right 3 units and up 2 units from f(x) = x2

Sketch the graphs of the following functions using horizontal and vertical shifting.

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 23 xy

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