Precalculus Examples

$y = {(x - 14)}^{2} - 9$

Step 1

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

$y = x^{2}$

Step 2

Simplify ${(x - 14)}^{2} - 9$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x - 14)}^{2}$ as $(x - 14) (x - 14)$ .

$y = (x - 14) (x - 14) - 9$

Step 2.1.2

Expand $(x - 14) (x - 14)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$y = x (x - 14) - 14 (x - 14) - 9$

Step 2.1.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot - 14 - 14 (x - 14) - 9$

Step 2.1.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14 - 9$

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 $x$ .

$y = x^{2} + x \cdot - 14 - 14 x - 14 \cdot - 14 - 9$

Step 2.1.3.1.2

Move $- 14$ to the left of $x$ .

$y = x^{2} - 14 \cdot x - 14 x - 14 \cdot - 14 - 9$

Step 2.1.3.1.3

Multiply $- 14$ by $- 14$ .

$y = x^{2} - 14 x - 14 x + 196 - 9$

Step 2.1.3.2

Subtract $14 x$ from $- 14 x$ .

$y = x^{2} - 28 x + 196 - 9$

Step 2.2

Subtract $9$ from $196$ .

$y = x^{2} - 28 x + 187$

Step 3

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $y = {(x - 14)}^{2} - 9$ is $g (x) = x^{2} - 28 x + 187$ .

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

$g (x) = x^{2} - 28 x + 187$

Step 4

The transformation being described is from $f (x) = x^{2}$ to $g (x) = x^{2} - 28 x + 187$ .

$f (x) = x^{2} \to g (x) = x^{2} - 28 x + 187$

Step 5

Find the vertex form of $g (x) = x^{2} - 28 x + 187$ .

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Step 5.1

Complete the square for $x^{2} - 28 x + 187$ .

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Step 5.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 28$

$c = 187$

Step 5.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 28}{2 \cdot 1}$

Step 5.1.3.2

Cancel the common factor of $- 28$ and $2$ .

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Step 5.1.3.2.1

Factor $2$ out of $- 28$ .

$d = \frac{2 \cdot - 14}{2 \cdot 1}$

Step 5.1.3.2.2

Cancel the common factors.

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Step 5.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 14}{2 (1)}$

Step 5.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 14}{2 \cdot 1}$

Step 5.1.3.2.2.3

Rewrite the expression.

$d = \frac{- 14}{1}$

Step 5.1.3.2.2.4

Divide $- 14$ by $1$ .

$d = - 14$

Step 5.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 187 - \frac{{(- 28)}^{2}}{4 \cdot 1}$

Step 5.1.4.2

Simplify the right side.

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Step 5.1.4.2.1

Simplify each term.

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Step 5.1.4.2.1.1

Raise $- 28$ to the power of $2$ .

$e = 187 - \frac{784}{4 \cdot 1}$

Step 5.1.4.2.1.2

Multiply $4$ by $1$ .

$e = 187 - \frac{784}{4}$

Step 5.1.4.2.1.3

Divide $784$ by $4$ .

$e = 187 - 1 \cdot 196$

Step 5.1.4.2.1.4

Multiply $- 1$ by $196$ .

$e = 187 - 196$

Step 5.1.4.2.2

Subtract $196$ from $187$ .

$e = - 9$

Step 5.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 14)}^{2} - 9$ .

${(x - 14)}^{2} - 9$

Step 5.2

Set $y$ equal to the new right side.

$y = {(x - 14)}^{2} - 9$

Step 6

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 $14$ Units

Step 7

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 $9$ Units

Step 8

The graph is reflected about the x-axis when $g (x) = - f (x)$ .

Reflection about the x-axis: None

Step 9

The graph is reflected about the y-axis when $g (x) = f (- x)$ .

Reflection about the y-axis: None

Step 10

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 11

Compare and list the transformations.

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

Horizontal Shift: Right $14$ Units

Vertical Shift: Down $9$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 12