Algebra Examples

$y = \frac{1}{2} {(x + 4)}^{2} - 8$

Step 1

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

$y = x^{2}$

Step 2

Simplify $\frac{1}{2} \cdot {(x + 4)}^{2} - 8$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$y = \frac{1}{2} \cdot ((x + 4) (x + 4)) - 8$

Step 2.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$y = \frac{1}{2} \cdot (x (x + 4) + 4 (x + 4)) - 8$

Step 2.1.2.2

Apply the distributive property.

$y = \frac{1}{2} \cdot (x \cdot x + x \cdot 4 + 4 (x + 4)) - 8$

Step 2.1.2.3

Apply the distributive property.

$y = \frac{1}{2} \cdot (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) - 8$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$y = \frac{1}{2} \cdot (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) - 8$

Step 2.1.3.1.2

Move $4$ to the left of $x$ .

$y = \frac{1}{2} \cdot (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) - 8$

Step 2.1.3.1.3

Multiply $4$ by $4$ .

$y = \frac{1}{2} \cdot (x^{2} + 4 x + 4 x + 16) - 8$

Step 2.1.3.2

Add $4 x$ and $4 x$ .

$y = \frac{1}{2} \cdot (x^{2} + 8 x + 16) - 8$

Step 2.1.4

Apply the distributive property.

$y = \frac{1}{2} x^{2} + \frac{1}{2} (8 x) + \frac{1}{2} \cdot 16 - 8$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y = \frac{x^{2}}{2} + \frac{1}{2} (8 x) + \frac{1}{2} \cdot 16 - 8$

Step 2.1.5.2

Cancel the common factor of $2$ .

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Step 2.1.5.2.1

Factor $2$ out of $8 x$ .

$y = \frac{x^{2}}{2} + \frac{1}{2} (2 (4 x)) + \frac{1}{2} \cdot 16 - 8$

Step 2.1.5.2.2

Cancel the common factor.

$y = \frac{x^{2}}{2} + \frac{1}{2} (2 (4 x)) + \frac{1}{2} \cdot 16 - 8$

Step 2.1.5.2.3

Rewrite the expression.

$y = \frac{x^{2}}{2} + 4 x + \frac{1}{2} \cdot 16 - 8$

Step 2.1.5.3

Cancel the common factor of $2$ .

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Step 2.1.5.3.1

Factor $2$ out of $16$ .

$y = \frac{x^{2}}{2} + 4 x + \frac{1}{2} \cdot (2 (8)) - 8$

Step 2.1.5.3.2

Cancel the common factor.

$y = \frac{x^{2}}{2} + 4 x + \frac{1}{2} \cdot (2 \cdot 8) - 8$

Step 2.1.5.3.3

Rewrite the expression.

$y = \frac{x^{2}}{2} + 4 x + 8 - 8$

Step 2.2

Combine the opposite terms in $\frac{x^{2}}{2} + 4 x + 8 - 8$ .

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Step 2.2.1

Subtract $8$ from $8$ .

$y = \frac{x^{2}}{2} + 4 x + 0$

Step 2.2.2

Add $\frac{x^{2}}{2} + 4 x$ and $0$ .

$y = \frac{x^{2}}{2} + 4 x$

Step 3

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $y = \frac{1}{2} \cdot {(x + 4)}^{2} - 8$ is $g (x) = \frac{x^{2}}{2} + 4 x$ .

$f (x) = x^{2}$

$g (x) = \frac{x^{2}}{2} + 4 x$

Step 4

The transformation being described is from $f (x) = x^{2}$ to $g (x) = \frac{x^{2}}{2} + 4 x$ .

$f (x) = x^{2} \to g (x) = \frac{x^{2}}{2} + 4 x$

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 $4$ 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 $8$ 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: 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: Compressed

Step 10

Compare and list the transformations.

Parent Function: $y = x^{2}$

Horizontal Shift: Left $4$ Units

Vertical Shift: Down $8$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: Compressed

Step 11