Precalculus Examples

$f (x) = 4 x - 4$ , $x = 2$

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.

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

Simplify the result.

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Apply the distributive property.

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

The final answer is $4 x + 4 h - 4$ .

$4 x + 4 h - 4$

Find the components of the definition.

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

$f (x) = 4 x - 4$

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

$f (x) = 4 x - 4$

Plug in the components.

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

Simplify.

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Simplify the numerator.

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Group the last three terms.

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

Factor.

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

Multiply $- 1$ by $4$ .

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

Remove parentheses.

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

Remove parentheses.

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

Factor $4$ out of $4 x + 4 h - 4 - 4 (x - 1)$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Remove parentheses.

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

Subtract $x$ from $x$ .

$\frac{4 (h - 1 + 0 + 1)}{h}$

Add $h$ and $0$ .

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

Add $- 1$ and $1$ .

$\frac{4 (h + 0)}{h}$

Add $h$ and $0$ .

$\frac{4 h}{h}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{4 h}{h}$

Divide $4$ by $1$ .

$4$

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