Precalculus Examples

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

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

Step 1

The parent function is the simplest form of the type of function given.

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

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

Step 2

Assume that

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

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

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

$f (x) = \frac{1}{x^{2}}$ and

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

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

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

$g (x) = \frac{1}{x^{2}}$ .

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

$f (x) = \frac{1}{x^{2}}$

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

$g (x) = \frac{1}{x^{2}}$

Step 3

The transformation from the first equation to the second one can be found by finding

a

$a$ ,

h

$h$ , and

k

$k$ for each equation.

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

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

Step 4

Find

a

$a$ ,

h

$h$ , and

k

$k$ for

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

$f (x) = \frac{1}{x^{2}}$ .

a = 1

$a = 1$

h = 0

$h = 0$

k = 0

$k = 0$

Step 5

Find

a

$a$ ,

h

$h$ , and

k

$k$ for

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

$g (x) = \frac{1}{x^{2}}$ .

a = 1

$a = 1$

h = 0

$h = 0$

k = 0

$k = 0$

Step 6

The horizontal shift depends on the value of

h

$h$ . The horizontal shift is described as:

g (x) = f (x + h)

$g (x) = f (x + h)$ - The graph is shifted to the left

h

$h$ units.

g (x) = f (x - h)

$g (x) = f (x - h)$ - The graph is shifted to the right

h

$h$ units.

Horizontal Shift: None

Step 7

The vertical shift depends on the value of

k

$k$ . The vertical shift is described as:

g (x) = f (x) + k

$g (x) = f (x) + k$ - The graph is shifted up

k

$k$ units.

g (x) = f (x) - k

$g (x) = f (x) - k$ - The graph is shifted down

k

$k$ units.

Vertical Shift: None

Step 8

The sign of

a

$a$ describes the reflection across the x-axis.

- a

$- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: None

Step 9

To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, and if there is a vertical stretch.

Parent Function:

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

$f (x) = \frac{1}{x^{2}}$

Horizontal Shift: None

Vertical Shift: None

Reflection about the x-axis: None

Step 10