Precalculus Examples

$f (x) = 12 x^{2} - 3$

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.

$f (x + h) = 12 {(x + h)}^{2} - 3$

Simplify the result.

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Simplify each term.

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

$f (x + h) = 12 ((x + h) (x + h)) - 3$

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

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Apply the distributive property.

$f (x + h) = 12 (x (x + h) + h (x + h)) - 3$

Apply the distributive property.

$f (x + h) = 12 (x x + x h + h (x + h)) - 3$

Apply the distributive property.

$f (x + h) = 12 (x x + x h + (h x + h h)) - 3$

Remove parentheses.

$f (x + h) = 12 (x x + x h + h x + h h) - 3$

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 $1$ .

$f (x + h) = 12 (x x + x h + h x + h h) - 3$

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

$f (x + h) = 12 (x x + x h + h x + h h) - 3$

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

$f (x + h) = 12 (x^{1 + 1} + x h + h x + h h) - 3$

Add $1$ and $1$ .

$f (x + h) = 12 (x^{2} + x h + h x + h h) - 3$

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 $1$ .

$f (x + h) = 12 (x^{2} + x h + h x + h h) - 3$

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

$f (x + h) = 12 (x^{2} + x h + h x + h h) - 3$

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

$f (x + h) = 12 (x^{2} + x h + h x + h^{1 + 1}) - 3$

Add $1$ and $1$ .

$f (x + h) = 12 (x^{2} + x h + h x + h^{2}) - 3$

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

$f (x + h) = 12 (x^{2} + (h x + h x) + h^{2}) - 3$

Add $h x$ and $h x$ .

$f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3$

Apply the distributive property.

$f (x + h) = 12 x^{2} + 12 (2 h x) + 12 h^{2} - 3$

Multiply $2$ by $12$ .

$f (x + h) = 12 x^{2} + 24 (h x) + 12 h^{2} - 3$

Remove parentheses.

$f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3$

The final answer is $12 x^{2} + 24 h x + 12 h^{2} - 3$ .

$12 x^{2} + 24 h x + 12 h^{2} - 3$

Find the components of the definition.

$f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3$

$f (x) = 12 x^{2} - 3$

$f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3$

$f (x) = 12 x^{2} - 3$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{12 x^{2} + 24 h x + 12 h^{2} - 3 - (12 x^{2} - 3)}{h}$

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$\frac{12 x^{2} + 24 h x + 12 h^{2} - 3 + (- (12 x^{2}) - - 3)}{h}$

Multiply $12$ by $- 1$ .

$\frac{12 x^{2} + 24 h x + 12 h^{2} - 3 + (- 12 x^{2} - - 3)}{h}$

Multiply $- 1$ by $- 3$ .

$\frac{12 x^{2} + 24 h x + 12 h^{2} - 3 + (- 12 x^{2} + 3)}{h}$

Remove unnecessary parentheses.

$\frac{12 x^{2} + 24 h x + 12 h^{2} - 3 - 12 x^{2} + 3}{h}$

Subtract $12 x^{2}$ from $12 x^{2}$ .

$\frac{24 h x + 12 h^{2} - 3 + 0 + 3}{h}$

Add $24 h x$ and $0$ .

$\frac{24 h x + 12 h^{2} - 3 + 3}{h}$

Add $- 3$ and $3$ .

$\frac{24 h x + 12 h^{2} + 0}{h}$

Add $24 h x + 12 h^{2}$ and $0$ .

$\frac{24 h x + 12 h^{2}}{h}$

Factor $12 h$ out of $24 h x + 12 h^{2}$ .

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Factor $12 h$ out of $24 h x$ .

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

Factor $12 h$ out of $12 h^{2}$ .

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

Factor $12 h$ out of $12 h (2 x) + 12 h h$ .

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

Simplify terms.

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Reduce the expression by cancelling the common factors.

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Cancel the common factor.

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

Divide $12 (2 x + h)$ by $1$ .

$12 (2 x + h)$

Apply the distributive property.

$12 (2 x) + 12 h$

Multiply $2$ by $12$ .

$24 x + 12 h$

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