Unit 3

Transformations

Invariant Points

An invariant point is one that does not change under a transformation.

In Example 1, the

x

-axis was the reflection line, and the

x

-intercept was an invariant point. The point

(- 1, 0)

stayed at

(- 1, 0)

after the transformation was applied.

In Example 2, the reflection line was the

y

-axis, and the

y

-intercept was an invariant point. The point

(0, 2)

stayed at

(0, 2)

after the transformation was applied.

Stretches

A stretch is just as it sounds. The shape of the graph changes, but the orientation of the graph and the basic position do not.

In the case of a vertical stretch about the

x

-axis, the

y

-values change by a common factor. If

y = f (x)

is the equation of the original function, then

y = a f (x)

is the equation of the function after a vertical stretch. All the

y

-values are multiplied by the factor

|a|

.

A horizontal stretch about the

y

-axis affects the

x

-values of the original function. If

y = f (x)

is the equation of the original function, then

y = f (b x)

is the equation of the function after a horizontal stretch. The

x

-values are all multiplied by the factor

\frac{1}{|b|}

.

stretch

Example 3

Given the function

f (x) = x^{2}

,

i.

sketch the graph of the transformed function $g (x)$ , for each case below,

ii.

describe the transformation involved in each case,

iii.

state any invariant points, and

iv.

state the domain and range of each transformed function.

a.

g (x) = 4 f (x)

b.

g (x) = (\frac{1}{3}) f (x)

c.

g (x) = f (4 x)

d.

g (x) = f (\frac{1}{3} x)

a.

i.

graph

ii.

In the transformed graph,

g (x)

, the

y

-values have been multiplied by a factor of

4

. This means each point on the graph of

g (x)

is

4

times as far from the

x

-axis as the corresponding point on

f (x)

. This gives the appearance of the graph being “taller”.
Note: It is tempting to think of

g (x)

as being “narrower” than

f (x)

. However, because we are interpreting the stretch as vertical, it is best to imagine individual points moving upwards instead of inwards.

iii.

The point

(0, 0)

is invariant.

iv.

The domain is the same for both functions.

\{x ∣ x \in R\}

The range is the same for both functions.

\{y ∣ y \geq 0, y \in R\}

b.

i.

graph

ii.

In the transformed graph,

g (x)

, the

y

-values have been multiplied by a factor of

\frac{1}{3}

. This means each point on the graph of

g (x)

is

\frac{1}{3}

as far from the

x

-axis as the corresponding point on

f (x)

. This gives the appearance of the graph being “shorter”.

iii.

The point

(0, 0)

is invariant.

iv.

The domain is the same for both functions.

\{x ∣ x \in R\}

The range is the same for both functions.

\{y ∣ y \geq 0, y \in R\}

c.

i.

graph

ii.

In the transformed graph,

g (x)

, the

x

-values have been multiplied by a factor of

\frac{1}{4}

. This means each point on the graph of

g (x)

is

\frac{1}{4}

as far from the

y

-axis as the corresponding point on

f (x)

. This gives the appearance of the graph being “narrower”.

iii.

The point

(0, 0)

is invariant.

iv.

The domain is the same for both functions.

\{x ∣ x \in R\}

The range is the same for both functions.

\{y ∣ y \geq 0, y \in R\}

d.

i.

graph

ii.

In the transformed graph,

g (x)

, the

x

-values have been multiplied by a factor of

3

. This means each point on the graph of

g (x)

is

3

times as far from the

y

-axis as the corresponding point on

f (x)

. This gives the appearance of the graph being “wider”.

iii.

The point

(0, 0)

is invariant.

iv.

The domain is the same for both functions.

\{x ∣ x \in R\}

The range is the same for both functions.

\{y ∣ y \geq 0, y \in R\}