Unit 3

Transformations

textbook

Part 3.1C corresponds to section 1.3, starting on page 32 of your Pre-Calculus 12 textbook.

You may have noticed the order in which a reflection and a stretch are applied does not affect the end result.

This is not true for all transformations.

The order may make a difference. To avoid errors, stretches and reflections should be done first, followed by translations.

Example 1

The graph of

f (x)

is shown.

a.

Describe the combination of transformations that must be applied to the function

y = f (x)

to obtain the transformed function

g (x)

, if

g (x) = 2 f (x) - 3

.

b.

Sketch the graph of

g (x)

on the same grid.

a.

The graph of

y = f (x)

is vertically stretched about the

x

-axisby a factor of

2

, and then translated

3

units down.

b.

To sketch the graph of

y = g (x)

, each point on the graph of

y = f (x)

is vertically stretched about the

x

-axis by a factor of

2

, and then translated

3

units down.

After the vertical stretch by a factor of

2

, the points

(0, 0), (1, 1), (4, 2),

and

(9, 3)

become

(0, 0), (1, 2), (4, 4),

and

(9, 6)

, respectively.

After the translation

3

units down, the points

(0, 0), (1, 2), (4, 4),

and

(9, 6)

become

(0, - 3), (1, - 1), (4, 1),

and

(9, 3)

, respectively.

Example 2

Describe the combination of transformations that should be applied to the graph of the function

f (x) = x^{2}

to obtain the graph of

g (x) = - 2 f (\frac{1}{2} (x + 8)) - 3

. Then, sketch the graphs of

f (x)

and

g (x)

on the same grid.

g (x) = a f (b (x - h)) + k

\begin{array}{rcl} |a| & = & 2 \\ |b| & = & \frac{1}{2} \\ h & = & - 8 \\ k & = & - 3 \end{array}

There is a vertical stretch about the

x

-axis by a factor of

2

because

|a| = 2

.

2 {(x)}^{2}

There is a horizontal stretch about the

y

-axis by a factor of

2

because

|b| = \frac{1}{2}

.

2 {(\frac{1}{2} x)}^{2}

There is a reflection in the

x

-axis because the value of

a

is negative.

- 2 {(\frac{1}{2} x)}^{2}

There is a horizontal translation

8

units left because

h = - 8

.

- 2 {(\frac{1}{2} (x + 8))}^{2}

There is a vertical translation

3

units down because

k = - 3

.

g (x) = - 2 {(\frac{1}{2} (x + 8))}^{2} - 3

Here are the graphs of both functions.

To sketch the graph of

y = g (x)

,

3

key points on the graph of

y = f (x)

, and their image points on

y = g (x)

, were located.

For the vertical stretch about the

x

-axis by a factor of

2

, each

y

-value was multiplied by

2

.

\begin{array}{l} (0, 0) \to (0, 0) \\ (- 2, 4) \to (- 2, 8) \\ (2, 4) \to (2, 8) \end{array}

For the horizontal stretch about the

y

-axis by a factor of

2

, each

x

-value was multiplied by

2

.

\begin{array}{l} (0, 0) \to (0, 0) \\ (- 2, 8) \to (- 4, 8) \\ (2, 8) \to (4, 8) \end{array}

After the reflection in the

x

-axis, each

y

-value was given the opposite sign.

\begin{array}{l} (0, 0) \to (0, 0) \\ (- 4, 8) \to (- 4, - 8) \\ (4, 8) \to (4, - 8) \end{array}

After the horizontal translation

8

units left, each

x

-value was reduced by

8

.

\begin{array}{l} (0, 0) \to (- 8, 0) \\ (- 4, - 8) \to (- 12, - 8) \\ (4, - 8) \to (- 4, - 8) \end{array}

After the vertical translation

3

units down, each

y

-value was reduced by

3

.

\begin{array}{l} (- 8, 0) \to (- 8, - 3) \\ (- 12, - 8) \to (- 12, - 11) \\ (- 4, - 8) \to (- 4, - 11) \end{array}