Algebra Examples

$y = - {(- x)}^{3}$

Step 1

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

$y = x^{3}$

Step 2

Simplify $- {(- x)}^{3}$ .

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Step 2.1

Apply the product rule to $- x$ .

$y = - ({(- 1)}^{3} x^{3})$

Step 2.2

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 2.2.1

Move ${(- 1)}^{3}$ .

$y = {(- 1)}^{3} \cdot - 1 x^{3}$

Step 2.2.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 2.2.2.1

Raise $- 1$ to the power of $1$ .

$y = {(- 1)}^{3} \cdot {(- 1)}^{1} x^{3}$

Step 2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = {(- 1)}^{3 + 1} x^{3}$

Step 2.2.3

Add $3$ and $1$ .

$y = {(- 1)}^{4} x^{3}$

Step 2.3

Raise $- 1$ to the power of $4$ .

$y = 1 x^{3}$

Step 2.4

Multiply $x^{3}$ by $1$ .

$y = x^{3}$

Step 3

Assume that $y = x^{3}$ is $f (x) = x^{3}$ and $y = - {(- x)}^{3}$ is $g (x) = x^{3}$ .

$f (x) = x^{3}$

$g (x) = x^{3}$

Step 4

The transformation being described is from $f (x) = x^{3}$ to $g (x) = x^{3}$ .

$f (x) = x^{3} \to g (x) = x^{3}$

Step 5

The horizontal shift depends on the value of $h$ . The horizontal shift is described as:

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

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

In this case, $h = 0$ which means that the graph is not shifted to the left or right.

Horizontal Shift: None

Step 6

The vertical shift depends on the value of $k$ . The vertical shift is described as:

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

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

In this case, $k = 0$ which means that the graph is not shifted up or down.

Vertical Shift: None

Step 7

The graph is reflected about the x-axis when $g (x) = - f (x)$ .

Reflection about the x-axis: None

Step 8

The graph is reflected about the y-axis when $g (x) = f (- x)$ .

Reflection about the y-axis: None

Step 9

Compressing and stretching depends on the value of $a$ .

When $a$ is greater than $1$ : Vertically stretched

When $a$ is between $0$ and $1$ : Vertically compressed

Vertical Compression or Stretch: None

Step 10

Compare and list the transformations.

Parent Function: $y = x^{3}$

Horizontal Shift: None

Vertical Shift: None

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 11