Precalculus Examples

$f (x) = 2 x^{2} + 12 x + 16$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 2 {(x + h)}^{2} + 12 (x + h) + 16$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

$f (x + h) = 2 ((x + h) (x + h)) + 12 (x + h) + 16$

Step 2.1.2.1.2

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

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 2 (x (x + h) + h (x + h)) + 12 (x + h) + 16$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h (x + h)) + 12 (x + h) + 16$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 2 (x \cdot x + x h + h x + h \cdot h) + 12 (x + h) + 16$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 2 (x^{2} + x h + h x + h \cdot h) + 12 (x + h) + 16$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = 2 (x^{2} + x h + h x + h^{2}) + 12 (x + h) + 16$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = 2 (x^{2} + h x + h x + h^{2}) + 12 (x + h) + 16$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = 2 (x^{2} + 2 h x + h^{2}) + 12 (x + h) + 16$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = 2 x^{2} + 2 (2 h x) + 2 h^{2} + 12 (x + h) + 16$

Step 2.1.2.1.5

Multiply $2$ by $2$ .

$f (x + h) = 2 x^{2} + 4 (h x) + 2 h^{2} + 12 (x + h) + 16$

Step 2.1.2.1.6

Apply the distributive property.

$f (x + h) = 2 x^{2} + 4 h x + 2 h^{2} + 12 x + 12 h + 16$

Step 2.1.2.2

The final answer is $2 x^{2} + 4 h x + 2 h^{2} + 12 x + 12 h + 16$ .

$2 x^{2} + 4 h x + 2 h^{2} + 12 x + 12 h + 16$

Step 2.2

Reorder.

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Step 2.2.1

Move $12 x$ .

$2 x^{2} + 4 h x + 2 h^{2} + 12 h + 12 x + 16$

Step 2.2.2

Move $2 x^{2}$ .

$4 h x + 2 h^{2} + 2 x^{2} + 12 h + 12 x + 16$

Step 2.2.3

Reorder $4 h x$ and $2 h^{2}$ .

$2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16$

Step 2.3

Find the components of the definition.

$f (x + h) = 2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16$

$f (x) = 2 x^{2} + 12 x + 16$

$f (x + h) = 2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16$

$f (x) = 2 x^{2} + 12 x + 16$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16 - (2 x^{2} + 12 x + 16)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16 - (2 x^{2}) - (12 x) - 1 \cdot 16}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $2$ by $- 1$ .

$\frac{2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16 - 2 x^{2} - (12 x) - 1 \cdot 16}{h}$

Step 4.1.2.2

Multiply $12$ by $- 1$ .

$\frac{2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16 - 2 x^{2} - 12 x - 1 \cdot 16}{h}$

Step 4.1.2.3

Multiply $- 1$ by $16$ .

$\frac{2 h^{2} + 4 h x + 2 x^{2} + 12 h + 12 x + 16 - 2 x^{2} - 12 x - 16}{h}$

Step 4.1.3

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{2 h^{2} + 4 h x + 12 h + 12 x + 16 + 0 - 12 x - 16}{h}$

Step 4.1.4

Add $2 h^{2}$ and $0$ .

$\frac{2 h^{2} + 4 h x + 12 h + 12 x + 16 - 12 x - 16}{h}$

Step 4.1.5

Subtract $12 x$ from $12 x$ .

$\frac{2 h^{2} + 4 h x + 12 h + 0 + 16 - 16}{h}$

Step 4.1.6

Add $2 h^{2}$ and $0$ .

$\frac{2 h^{2} + 4 h x + 12 h + 16 - 16}{h}$

Step 4.1.7

Subtract $16$ from $16$ .

$\frac{2 h^{2} + 4 h x + 12 h + 0}{h}$

Step 4.1.8

Add $2 h^{2} + 4 h x + 12 h$ and $0$ .

$\frac{2 h^{2} + 4 h x + 12 h}{h}$

Step 4.1.9

Factor $2 h$ out of $2 h^{2} + 4 h x + 12 h$ .

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Step 4.1.9.1

Factor $2 h$ out of $2 h^{2}$ .

$\frac{2 h \cdot h + 4 h x + 12 h}{h}$

Step 4.1.9.2

Factor $2 h$ out of $4 h x$ .

$\frac{2 h \cdot h + 2 h (2 x) + 12 h}{h}$

Step 4.1.9.3

Factor $2 h$ out of $12 h$ .

$\frac{2 h \cdot h + 2 h (2 x) + 2 h \cdot 6}{h}$

Step 4.1.9.4

Factor $2 h$ out of $2 h \cdot h + 2 h (2 x)$ .

$\frac{2 h (h + 2 x) + 2 h \cdot 6}{h}$

Step 4.1.9.5

Factor $2 h$ out of $2 h (h + 2 x) + 2 h \cdot 6$ .

$\frac{2 h (h + 2 x + 6)}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 h (h + 2 x + 6)}{h}$

Step 4.2.1.2

Divide $2 (h + 2 x + 6)$ by $1$ .

$2 (h + 2 x + 6)$

Step 4.2.2

Apply the distributive property.

$2 h + 2 (2 x) + 2 \cdot 6$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $2$ by $2$ .

$2 h + 4 x + 2 \cdot 6$

Step 4.3.2

Multiply $2$ by $6$ .

$2 h + 4 x + 12$

Step 4.4

Reorder $2 h$ and $4 x$ .

$4 x + 2 h + 12$

Step 5