Unit 3

Transformations

The previous section dealt with rational functions with linear expressions in the numerator and the denominator. As such, we compared the rational function

y = \frac{1}{x}

to the transformed function of the form

y = \frac{a}{x - h} + k

.

Next, you will investigate the base rational function

y = \frac{1}{x^{2}}

.

Example 3

Graph each function, and discuss their characteristics.

a.

y = \frac{1}{x^{2}}

b.

y = - \frac{1}{{(x - 3)}^{2}}

c.

y = 2 + \frac{5}{x^{2} + 2 x + 1}

a.

The graph of

y = \frac{1}{x^{2}}

is shown in black. It has a vertical asymptote at

x = 0

and a horizontal asymptote at

y = 0

.

As the values of

x

approach the vertical asymptote from either side, the values of

y

approach infinity.

The domain is

\{x ∣ x \neq 0, x \in R\}

.

The range is

\{y ∣ y > 0, y \in R\}

.

b.

The graph of

y = - \frac{1}{{(x - 3)}^{2}}

is shown in red. Compared to the graph of

y = \frac{1}{x^{2}}

, there has been a reflection in the

x

-axis meaning all the

y

-values are the opposite sign. There has also been a horizontal translation

3

units to the right, so the vertical asymptote is now at

x = 3

.

The domain is

\{x ∣ x \neq 3, x \in R\}

.

The range is

\{y ∣ y < 0, y \in R\}

.

c.

The graph of

y = 2 + \frac{5}{x^{2} + 2 x + 1}

is shown in red. The equation can be written in factored form as

y = 2 + \frac{5}{{(x + 1)}^{2}}

or

y = \frac{5}{{(x + 1)}^{2}} + 2

.

Compared to the graph of

y = \frac{1}{x^{2}}

, there has been a vertical stretch about the

x

-axis by a factor of

5

, a horizontal translation

1

unit left, and a vertical translation

2

units up. The vertical asymptote is now at

x = - 1

and the horizontal asymptote is at

y = 2

.

The domain is

\{x ∣ x \neq - 1, x \in R\}

.

The range is

y \geq 2, y \in R

.

It is important to note that

\begin{array}{rcl} y & = & 2 + \frac{5}{x^{2} + 2 x + 1} \\ = & 2 + \frac{5}{{(x + 1)}^{2}} \end{array}

In general,

y = \frac{a}{{(x - h)}^{2}} + k

, compared to

y = \frac{1}{x^{2}}

, has a vertical stretch about the

x

-axis by a factor of

a

, and if

a

is negative, there is a reflection in the

x

-axis.

It has a horizontal translation of

h

and a vertical translation of

k

.