Precalculus Examples

$f (x) = - \frac{x^{2}}{4} + 7$

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.

$f (x + h) = - \frac{{(x + h)}^{2}}{4} + 7$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (x + h) = - \frac{{(x + h)}^{2}}{4} + 7 \cdot \frac{4}{4}$

Step 2.1.2.2

Simplify terms.

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Step 2.1.2.2.1

Combine $7$ and $\frac{4}{4}$ .

$f (x + h) = - \frac{{(x + h)}^{2}}{4} + \frac{7 \cdot 4}{4}$

Step 2.1.2.2.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{- {(x + h)}^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3

Simplify the numerator.

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Step 2.1.2.3.1

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

$f (x + h) = \frac{- ((x + h) (x + h)) + 7 \cdot 4}{4}$

Step 2.1.2.3.2

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

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Step 2.1.2.3.2.1

Apply the distributive property.

$f (x + h) = \frac{- (x (x + h) + h (x + h)) + 7 \cdot 4}{4}$

Step 2.1.2.3.2.2

Apply the distributive property.

$f (x + h) = \frac{- (x \cdot x + x h + h (x + h)) + 7 \cdot 4}{4}$

Step 2.1.2.3.2.3

Apply the distributive property.

$f (x + h) = \frac{- (x \cdot x + x h + h x + h \cdot h) + 7 \cdot 4}{4}$

Step 2.1.2.3.3

Simplify and combine like terms.

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Step 2.1.2.3.3.1

Simplify each term.

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Step 2.1.2.3.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = \frac{- (x^{2} + x h + h x + h \cdot h) + 7 \cdot 4}{4}$

Step 2.1.2.3.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = \frac{- (x^{2} + x h + h x + h^{2}) + 7 \cdot 4}{4}$

Step 2.1.2.3.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = \frac{- (x^{2} + h x + h x + h^{2}) + 7 \cdot 4}{4}$

Step 2.1.2.3.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = \frac{- (x^{2} + 2 h x + h^{2}) + 7 \cdot 4}{4}$

Step 2.1.2.3.4

Apply the distributive property.

$f (x + h) = \frac{- x^{2} - (2 h x) - h^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3.5

Multiply $2$ by $- 1$ .

$f (x + h) = \frac{- x^{2} - 2 (h x) - h^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3.6

Multiply $7$ by $4$ .

$f (x + h) = \frac{- x^{2} - 2 h x - h^{2} + 28}{4}$

Step 2.1.2.4

Simplify with factoring out.

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Step 2.1.2.4.1

Factor $- 1$ out of $- x^{2}$ .

$f (x + h) = \frac{- (x^{2}) - 2 h x - h^{2} + 28}{4}$

Step 2.1.2.4.2

Factor $- 1$ out of $- 2 h x$ .

$f (x + h) = \frac{- (x^{2}) - (2 h x) - h^{2} + 28}{4}$

Step 2.1.2.4.3

Factor $- 1$ out of $- (x^{2}) - (2 h x)$ .

$f (x + h) = \frac{- (x^{2} + 2 h x) - h^{2} + 28}{4}$

Step 2.1.2.4.4

Factor $- 1$ out of $- h^{2}$ .

$f (x + h) = \frac{- (x^{2} + 2 h x) - (h^{2}) + 28}{4}$

Step 2.1.2.4.5

Factor $- 1$ out of $- (x^{2} + 2 h x) - (h^{2})$ .

$f (x + h) = \frac{- (x^{2} + 2 h x + h^{2}) + 28}{4}$

Step 2.1.2.4.6

Rewrite $28$ as $- 1 (- 28)$ .

$f (x + h) = \frac{- (x^{2} + 2 h x + h^{2}) - 1 \cdot - 28}{4}$

Step 2.1.2.4.7

Factor $- 1$ out of $- (x^{2} + 2 h x + h^{2}) - 1 (- 28)$ .

$f (x + h) = \frac{- (x^{2} + 2 h x + h^{2} - 28)}{4}$

Step 2.1.2.4.8

Simplify the expression.

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Step 2.1.2.4.8.1

Rewrite $- (x^{2} + 2 h x + h^{2} - 28)$ as $- 1 (x^{2} + 2 h x + h^{2} - 28)$ .

$f (x + h) = \frac{- 1 (x^{2} + 2 h x + h^{2} - 28)}{4}$

Step 2.1.2.4.8.2

Move the negative in front of the fraction.

$f (x + h) = - \frac{x^{2} + 2 h x + h^{2} - 28}{4}$

Step 2.1.2.5

The final answer is $- \frac{x^{2} + 2 h x + h^{2} - 28}{4}$ .

$- \frac{x^{2} + 2 h x + h^{2} - 28}{4}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{x^{2} + 2 h x + h^{2} - 28}{4}$

$f (x) = - \frac{x^{2}}{4} + 7$

$f (x + h) = - \frac{x^{2} + 2 h x + h^{2} - 28}{4}$

$f (x) = - \frac{x^{2}}{4} + 7$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{x^{2} + 2 h x + h^{2} - 28}{4} - (- \frac{x^{2}}{4} + 7)}{h}$

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.

$\frac{- \frac{x^{2} + 2 h x + h^{2} - 28}{4} - - \frac{x^{2}}{4} - 1 \cdot 7}{h}$

Step 4.1.2

Multiply $- - \frac{x^{2}}{4}$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{x^{2} + 2 h x + h^{2} - 28}{4} + 1 \frac{x^{2}}{4} - 1 \cdot 7}{h}$

Step 4.1.2.2

Multiply $\frac{x^{2}}{4}$ by $1$ .

$\frac{- \frac{x^{2} + 2 h x + h^{2} - 28}{4} + \frac{x^{2}}{4} - 1 \cdot 7}{h}$

Step 4.1.3

Multiply $- 1$ by $7$ .

$\frac{- \frac{x^{2} + 2 h x + h^{2} - 28}{4} + \frac{x^{2}}{4} - 7}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{- (x^{2} + 2 h x + h^{2} - 28) + x^{2}}{4} - 7}{h}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Apply the distributive property.

$\frac{\frac{- x^{2} - (2 h x) - h^{2} - - 28 + x^{2}}{4} - 7}{h}$

Step 4.1.5.2

Simplify.

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Step 4.1.5.2.1

Multiply $2$ by $- 1$ .

$\frac{\frac{- x^{2} - 2 (h x) - h^{2} - - 28 + x^{2}}{4} - 7}{h}$

Step 4.1.5.2.2

Multiply $- 1$ by $- 28$ .

$\frac{\frac{- x^{2} - 2 (h x) - h^{2} + 28 + x^{2}}{4} - 7}{h}$

Step 4.1.5.3

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

$\frac{\frac{- 2 h x - h^{2} + 28 + 0}{4} - 7}{h}$

Step 4.1.5.4

Add $- 2 h x - h^{2} + 28$ and $0$ .

$\frac{\frac{- 2 h x - h^{2} + 28}{4} - 7}{h}$

Step 4.1.6

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{- 2 h x - h^{2} + 28}{4} - 7 \cdot \frac{4}{4}}{h}$

Step 4.1.7

Combine $- 7$ and $\frac{4}{4}$ .

$\frac{\frac{- 2 h x - h^{2} + 28}{4} + \frac{- 7 \cdot 4}{4}}{h}$

Step 4.1.8

Combine the numerators over the common denominator.

$\frac{\frac{- 2 h x - h^{2} + 28 - 7 \cdot 4}{4}}{h}$

Step 4.1.9

Simplify the numerator.

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Step 4.1.9.1

Multiply $- 7$ by $4$ .

$\frac{\frac{- 2 h x - h^{2} + 28 - 28}{4}}{h}$

Step 4.1.9.2

Subtract $28$ from $28$ .

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

Step 4.1.9.3

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

$\frac{\frac{- 2 h x - h^{2}}{4}}{h}$

Step 4.1.9.4

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

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Step 4.1.9.4.1

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

$\frac{\frac{h (- 2 x) - h^{2}}{4}}{h}$

Step 4.1.9.4.2

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

$\frac{\frac{h (- 2 x) + h (- h)}{4}}{h}$

Step 4.1.9.4.3

Factor $h$ out of $h (- 2 x) + h (- h)$ .

$\frac{\frac{h (- 2 x - h)}{4}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{h (- 2 x - h)}{4} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

$\frac{h (- 2 x - h)}{4} \cdot \frac{1}{h}$

Step 4.3.2

Rewrite the expression.

$\frac{- 2 x - h}{4}$

Step 4.4

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) - h}{4}$

Step 4.5

Factor $- 1$ out of $- (2 x) - h$ .

$\frac{- (2 x + h)}{4}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Rewrite $- (2 x + h)$ as $- 1 (2 x + h)$ .

$\frac{- 1 (2 x + h)}{4}$

Step 4.6.2

Move the negative in front of the fraction.

$- \frac{2 x + h}{4}$

Step 5