Precalculus Examples

$f (x) = 3 x^{2} + 6$

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.

$f (x + h) = 3 {(x + h)}^{2} + 6$

Simplify the result.

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Simplify each term.

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

$f (x + h) = 3 ((x + h) (x + h)) + 6$

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

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Apply the distributive property.

$f (x + h) = 3 (x (x + h) + h (x + h)) + 6$

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h (x + h)) + 6$

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h x + h \cdot h) + 6$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$f (x + h) = 3 (x^{2} + x h + h x + h \cdot h) + 6$

Multiply $h$ by $h$ .

$f (x + h) = 3 (x^{2} + x h + h x + h^{2}) + 6$

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

$f (x + h) = 3 (x^{2} + (h x + h x) + h^{2}) + 6$

Add $h x$ and $h x$ .

$f (x + h) = 3 (x^{2} + 2 h x + h^{2}) + 6$

Apply the distributive property.

$f (x + h) = 3 x^{2} + 3 (2 h x) + 3 h^{2} + 6$

Multiply $2$ by $3$ .

$f (x + h) = 3 x^{2} + 6 h x + 3 h^{2} + 6$

The final answer is $3 x^{2} + 6 h x + 3 h^{2} + 6$ .

$3 x^{2} + 6 h x + 3 h^{2} + 6$

Find the components of the definition.

$f (x + h) = 3 x^{2} + 6 h x + 3 h^{2} + 6$

$f (x) = 3 x^{2} + 6$

$f (x + h) = 3 x^{2} + 6 h x + 3 h^{2} + 6$

$f (x) = 3 x^{2} + 6$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{3 x^{2} + 6 h x + 3 h^{2} + 6 - (3 x^{2} + 6)}{h}$

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$\frac{3 x^{2} + 6 h x + 3 h^{2} + 6 + (- (3 x^{2}) - 1 \cdot 6)}{h}$

Multiply $3$ by $- 1$ .

$\frac{3 x^{2} + 6 h x + 3 h^{2} + 6 + (- 3 x^{2} - 1 \cdot 6)}{h}$

Multiply $- 1$ by $6$ .

$\frac{3 x^{2} + 6 h x + 3 h^{2} + 6 + (- 3 x^{2} - 6)}{h}$

Subtract $3 x^{2}$ from $3 x^{2}$ .

$\frac{6 h x + 3 h^{2} + 6 + 0 - 6}{h}$

Add $6 h x$ and $0$ .

$\frac{6 h x + 3 h^{2} + 6 - 6}{h}$

Subtract $6$ from $6$ .

$\frac{6 h x + 3 h^{2} + 0}{h}$

Add $6 h x + 3 h^{2}$ and $0$ .

$\frac{6 h x + 3 h^{2}}{h}$

Factor $3 h$ out of $6 h x + 3 h^{2}$ .

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Factor $3 h$ out of $6 h x$ .

$\frac{3 h (2 x) + 3 h^{2}}{h}$

Factor $3 h$ out of $3 h^{2}$ .

$\frac{3 h (2 x) + 3 h \cdot h}{h}$

Factor $3 h$ out of $3 h (2 x) + 3 h \cdot h$ .

$\frac{3 h (2 x + h)}{h}$

Simplify terms.

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Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{3 h (2 x + h)}{h}$

Divide $3 (2 x + h)$ by $1$ .

$3 (2 x + h)$

Apply the distributive property.

$3 (2 x) + 3 h$

Multiply $2$ by $3$ .

$6 x + 3 h$

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