Algebra Examples

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

Step 1

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

$y = x^{3}$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Use the Binomial Theorem.

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

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

Multiply $3$ by $3$ .

$y = - (x^{3} + 9 x^{2} + 3 x \cdot 3^{2} + 3^{3}) - 5$

Step 2.1.2.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 2.1.2.2.1

Move $3^{2}$ .

$y = - (x^{3} + 9 x^{2} + 3^{2} \cdot 3 x + 3^{3}) - 5$

Step 2.1.2.2.2

Multiply $3^{2}$ by $3$ .

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Step 2.1.2.2.2.1

Raise $3$ to the power of $1$ .

$y = - (x^{3} + 9 x^{2} + 3^{2} \cdot 3^{1} x + 3^{3}) - 5$

Step 2.1.2.2.2.2

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

$y = - (x^{3} + 9 x^{2} + 3^{2 + 1} x + 3^{3}) - 5$

Step 2.1.2.2.3

Add $2$ and $1$ .

$y = - (x^{3} + 9 x^{2} + 3^{3} x + 3^{3}) - 5$

Step 2.1.2.3

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

$y = - (x^{3} + 9 x^{2} + 27 x + 3^{3}) - 5$

Step 2.1.2.4

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

$y = - (x^{3} + 9 x^{2} + 27 x + 27) - 5$

Step 2.1.3

Apply the distributive property.

$y = - x^{3} - (9 x^{2}) - (27 x) - 1 \cdot 27 - 5$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Multiply $9$ by $- 1$ .

$y = - x^{3} - 9 x^{2} - (27 x) - 1 \cdot 27 - 5$

Step 2.1.4.2

Multiply $27$ by $- 1$ .

$y = - x^{3} - 9 x^{2} - 27 x - 1 \cdot 27 - 5$

Step 2.1.4.3

Multiply $- 1$ by $27$ .

$y = - x^{3} - 9 x^{2} - 27 x - 27 - 5$

Step 2.2

Subtract $5$ from $- 27$ .

$y = - x^{3} - 9 x^{2} - 27 x - 32$

Step 3

Assume that $y = x^{3}$ is $f (x) = x^{3}$ and $y = - {(x + 3)}^{3} - 5$ is $g (x) = - x^{3} - 9 x^{2} - 27 x - 32$ .

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

$g (x) = - x^{3} - 9 x^{2} - 27 x - 32$

Step 4

The transformation being described is from $f (x) = x^{3}$ to $g (x) = - x^{3} - 9 x^{2} - 27 x - 32$ .

$f (x) = x^{3} \to g (x) = - x^{3} - 9 x^{2} - 27 x - 32$

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 $3$ 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.

Vertical Shift: Down $5$ Units

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: None

Step 10

Compare and list the transformations.

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

Horizontal Shift: Left $3$ Units

Vertical Shift: Down $5$ Units

Reflection about the x-axis: None

Reflection about the y-axis: Reflected

Vertical Compression or Stretch: None

Step 11