Precalculus Examples

$f (x) = x^{2} + 8 x - 4$ , $x = 4$

Consider the difference quotient formula. Use this formula to solve for the slope of the secant line $m_{s e c}$ .

$m_{s e c} = \frac{f (x + h) - f (x)}{h}$

Find the components of the definition.

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

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

$f (x + h) = {(x + h)}^{2} + 8 (x + h) - 4$

Simplify the result.

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Simplify each term.

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

$f (x + h) = (x + h) (x + h) + 8 (x + h) - 4$

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

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Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) + 8 (x + h) - 4$

Apply the distributive property.

$f (x + h) = x x + x h + h (x + h) + 8 (x + h) - 4$

Apply the distributive property.

$f (x + h) = x x + x h + (h x + h h) + 8 (x + h) - 4$

Remove parentheses.

$f (x + h) = x x + x h + h x + h h + 8 (x + h) - 4$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ by adding the exponents.

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Combine $x$ and $x$

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Raise $x$ to the power of $1$ .

$f (x + h) = x x + x h + h x + h h + 8 (x + h) - 4$

Raise $x$ to the power of $1$ .

$f (x + h) = x x + x h + h x + h h + 8 (x + h) - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x + h) = x^{1 + 1} + x h + h x + h h + 8 (x + h) - 4$

Add $1$ and $1$ .

$f (x + h) = x^{2} + x h + h x + h h + 8 (x + h) - 4$

Multiply $h$ by $h$ by adding the exponents.

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Combine $h$ and $h$

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Raise $h$ to the power of $1$ .

$f (x + h) = x^{2} + x h + h x + h h + 8 (x + h) - 4$

Raise $h$ to the power of $1$ .

$f (x + h) = x^{2} + x h + h x + h h + 8 (x + h) - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x + h) = x^{2} + x h + h x + h^{1 + 1} + 8 (x + h) - 4$

Add $1$ and $1$ .

$f (x + h) = x^{2} + x h + h x + h^{2} + 8 (x + h) - 4$

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

$f (x + h) = x^{2} + (h x + h x) + h^{2} + 8 (x + h) - 4$

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 8 (x + h) - 4$

Apply the distributive property.

$f (x + h) = x^{2} + 2 h x + h^{2} + (8 x + 8 h) - 4$

Remove unnecessary parentheses.

$f (x + h) = x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4$

The final answer is $x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4$ .

$x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4$

Find the components of the definition.

$f (x + h) = x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4$

$f (x) = x^{2} + 8 x - 4$

$f (x + h) = x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4$

$f (x) = x^{2} + 8 x - 4$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4 - (x^{2} + 8 x - 4)}{h}$

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$\frac{x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4 + (- x^{2} - (8 x) - - 4)}{h}$

Simplify.

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Multiply $8$ by $- 1$ .

$\frac{x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4 + (- x^{2} - 8 x - - 4)}{h}$

Multiply $- 1$ by $- 4$ .

$\frac{x^{2} + 2 h x + h^{2} + 8 x + 8 h - 4 + (- x^{2} - 8 x + 4)}{h}$

Remove unnecessary parentheses.

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

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

$\frac{2 h x + h^{2} + 8 x + 8 h - 4 + 0 - 8 x + 4}{h}$

Add $2 h x$ and $0$ .

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

Subtract $8 x$ from $8 x$ .

$\frac{2 h x + h^{2} + 8 h - 4 + 0 + 4}{h}$

Add $2 h x$ and $0$ .

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

Add $- 4$ and $4$ .

$\frac{2 h x + h^{2} + 8 h + 0}{h}$

Add $2 h x + h^{2} + 8 h$ and $0$ .

$\frac{2 h x + h^{2} + 8 h}{h}$

Factor $h$ out of $2 h x + h^{2} + 8 h$ .

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

$\frac{h (2 x) + h^{2} + 8 h}{h}$

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

$\frac{h (2 x) + h h + 8 h}{h}$

Factor $h$ out of $8 h$ .

$\frac{h (2 x) + h h + h \cdot 8}{h}$

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

$\frac{h (2 x + h) + h \cdot 8}{h}$

Factor $h$ out of $h (2 x + h) + h \cdot 8$ .

$\frac{h (2 x + h + 8)}{h}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{h (2 x + h + 8)}{h}$

Divide $2 x + h + 8$ by $1$ .

$2 x + h + 8$

Replace the variable $x$ with $4$ in the expression.

$2 (4) + h + 8$

Multiply $2$ by $4$ .

$8 + h + 8$

Add $8$ and $8$ .

$h + 16$

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