Algebra Examples

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

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

Step 1

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

y = x^{3}

$y = x^{3}$

Step 2

Simplify

- 2 {(x + 5)}^{3}

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

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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})

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

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

$5$ by

3

$3$ .

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

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

Step 2.2.1.2

Raise

5

$5$ to the power of

2

$2$ .

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

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

Step 2.2.1.3

Multiply

25

$25$ by

3

$3$ .

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

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

Step 2.2.1.4

Raise

5

$5$ to the power of

3

$3$ .

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

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

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

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

Step 2.2.2

Apply the distributive property.

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

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

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

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

Step 2.3

Simplify.

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Step 2.3.1

Multiply

15

$15$ by

- 2

$- 2$ .

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

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

Step 2.3.2

Multiply

75

$75$ by

- 2

$- 2$ .

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

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

Step 2.3.3

Multiply

- 2

$- 2$ by

125

$125$ .

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

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

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

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

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

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

Step 3

Assume that

y = x^{3}

$y = x^{3}$ is

f (x) = x^{3}

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

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

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

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

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

f (x) = x^{3}

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

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

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

Step 4

The transformation being described is from

f (x) = x^{3}

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

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

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

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

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

Step 5

The horizontal shift depends on the value of

h

$h$ . 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: Left

5

$5$ Units

Step 6

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.

In this case,

k = 0

$k = 0$ 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)

$g (x) = - f (x)$ .

Reflection about the x-axis: None

Step 8

The graph is reflected about the y-axis when

g (x) = f (- x)

$g (x) = f (- x)$ .

Reflection about the y-axis: Reflected

Step 9

Compressing and stretching depends on the value of

a

$a$ .

When

a

$a$ is greater than

1

$1$ : Vertically stretched

When

a

$a$ is between

0

$0$ and

1

$1$ : Vertically compressed

Vertical Compression or Stretch: Stretched

Step 10

Compare and list the transformations.

Parent Function:

y = x^{3}

$y = x^{3}$

Horizontal Shift: Left

5

$5$ Units

Vertical Shift: None

Reflection about the x-axis: None

Reflection about the y-axis: Reflected

Vertical Compression or Stretch: Stretched

Step 11