Algebra Examples

$y = - 2 {(x + 5)}^{3}$

Step 1

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

$y = x^{3}$

Step 2

Simplify $- 2 {(x + 5)}^{3}$ .

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Step 2.1

Use the Binomial Theorem.

$y = - 2 (x^{3} + 3 x^{2} \cdot 5 + 3 x \cdot 5^{2} + 5^{3})$

Step 2.2

Simplify terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $5$ by $3$ .

$y = - 2 (x^{3} + 15 x^{2} + 3 x \cdot 5^{2} + 5^{3})$

Step 2.2.1.2

Raise $5$ to the power of $2$ .

$y = - 2 (x^{3} + 15 x^{2} + 3 x \cdot 25 + 5^{3})$

Step 2.2.1.3

Multiply $25$ by $3$ .

$y = - 2 (x^{3} + 15 x^{2} + 75 x + 5^{3})$

Step 2.2.1.4

Raise $5$ to the power of $3$ .

$y = - 2 (x^{3} + 15 x^{2} + 75 x + 125)$

Step 2.2.2

Apply the distributive property.

$y = - 2 x^{3} - 2 (15 x^{2}) - 2 (75 x) - 2 \cdot 125$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $15$ by $- 2$ .

$y = - 2 x^{3} - 30 x^{2} - 2 (75 x) - 2 \cdot 125$

Step 2.3.2

Multiply $75$ by $- 2$ .

$y = - 2 x^{3} - 30 x^{2} - 150 x - 2 \cdot 125$

Step 2.3.3

Multiply $- 2$ by $125$ .

$y = - 2 x^{3} - 30 x^{2} - 150 x - 250$

Step 3

Assume that $y = x^{3}$ is $f (x) = x^{3}$ and $y = - 2 {(x + 5)}^{3}$ is $g (x) = - 2 x^{3} - 30 x^{2} - 150 x - 250$ .

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

$g (x) = - 2 x^{3} - 30 x^{2} - 150 x - 250$

Step 4

The transformation being described is from $f (x) = x^{3}$ to $g (x) = - 2 x^{3} - 30 x^{2} - 150 x - 250$ .

$f (x) = x^{3} \to g (x) = - 2 x^{3} - 30 x^{2} - 150 x - 250$

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.

Horizontal Shift: Left $5$ Units

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: Reflected

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: Stretched

Step 10

Compare and list the transformations.

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

Horizontal Shift: Left $5$ Units

Vertical Shift: None

Reflection about the x-axis: None

Reflection about the y-axis: Reflected

Vertical Compression or Stretch: Stretched

Step 11