Algebra Examples

$y = {(x - 2)}^{2} - 4$

Step 1

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

$y = x^{2}$

Step 2

Simplify ${(x - 2)}^{2} - 4$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$y = (x - 2) (x - 2) - 4$

Step 2.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$y = x (x - 2) - 2 (x - 2) - 4$

Step 2.1.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 2 - 2 (x - 2) - 4$

Step 2.1.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 4$

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 $x$ .

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

Step 2.1.3.1.2

Move $- 2$ to the left of $x$ .

$y = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 4$

Step 2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$y = x^{2} - 2 x - 2 x + 4 - 4$

Step 2.1.3.2

Subtract $2 x$ from $- 2 x$ .

$y = x^{2} - 4 x + 4 - 4$

Step 2.2

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

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Step 2.2.1

Subtract $4$ from $4$ .

$y = x^{2} - 4 x + 0$

Step 2.2.2

Add $x^{2} - 4 x$ and $0$ .

$y = x^{2} - 4 x$

Step 3

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $y = {(x - 2)}^{2} - 4$ is $g (x) = x^{2} - 4 x$ .

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

$g (x) = x^{2} - 4 x$

Step 4

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

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

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: Right $2$ 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 $4$ 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: None

Step 10

Compare and list the transformations.

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

Horizontal Shift: Right $2$ Units

Vertical Shift: Down $4$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 11