Precalculus Examples

$y = \frac{1}{3} x^{2} - 4$

Step 1

Write $y = \frac{1}{3} x^{2} - 4$ as a function.

$f (x) = \frac{1}{3} x^{2} - 4$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = \frac{1}{3} \cdot {(x + h)}^{2} - 4$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$f (x + h) = \frac{1}{3} \cdot ((x + h) (x + h)) - 4$

Step 3.1.2.1.2

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

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Step 3.1.2.1.2.1

Apply the distributive property.

$f (x + h) = \frac{1}{3} \cdot (x (x + h) + h (x + h)) - 4$

Step 3.1.2.1.2.2

Apply the distributive property.

$f (x + h) = \frac{1}{3} \cdot (x \cdot x + x h + h (x + h)) - 4$

Step 3.1.2.1.2.3

Apply the distributive property.

$f (x + h) = \frac{1}{3} \cdot (x \cdot x + x h + h x + h \cdot h) - 4$

Step 3.1.2.1.3

Simplify and combine like terms.

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Step 3.1.2.1.3.1

Simplify each term.

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Step 3.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = \frac{1}{3} \cdot (x^{2} + x h + h x + h \cdot h) - 4$

Step 3.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = \frac{1}{3} \cdot (x^{2} + x h + h x + h^{2}) - 4$

Step 3.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 3.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = \frac{1}{3} \cdot (x^{2} + h x + h x + h^{2}) - 4$

Step 3.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = \frac{1}{3} \cdot (x^{2} + 2 h x + h^{2}) - 4$

Step 3.1.2.1.4

Apply the distributive property.

$f (x + h) = \frac{1}{3} x^{2} + \frac{1}{3} \cdot (2 h x) + \frac{1}{3} h^{2} - 4$

Step 3.1.2.1.5

Simplify.

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Step 3.1.2.1.5.1

Combine $\frac{1}{3}$ and $x^{2}$ .

$f (x + h) = \frac{x^{2}}{3} + \frac{1}{3} \cdot (2 h x) + \frac{1}{3} h^{2} - 4$

Step 3.1.2.1.5.2

Multiply $\frac{1}{3} (2 h x)$ .

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Step 3.1.2.1.5.2.1

Combine $2$ and $\frac{1}{3}$ .

$f (x + h) = \frac{x^{2}}{3} + \frac{2}{3} \cdot (h x) + \frac{1}{3} h^{2} - 4$

Step 3.1.2.1.5.2.2

Combine $h$ and $\frac{2}{3}$ .

$f (x + h) = \frac{x^{2}}{3} + \frac{h \cdot 2}{3} \cdot x + \frac{1}{3} h^{2} - 4$

Step 3.1.2.1.5.2.3

Combine $\frac{h \cdot 2}{3}$ and $x$ .

$f (x + h) = \frac{x^{2}}{3} + \frac{h \cdot (2 x)}{3} + \frac{1}{3} h^{2} - 4$

Step 3.1.2.1.5.3

Combine $\frac{1}{3}$ and $h^{2}$ .

$f (x + h) = \frac{x^{2}}{3} + \frac{h \cdot (2 x)}{3} + \frac{h^{2}}{3} - 4$

Step 3.1.2.1.6

Move $2$ to the left of $h$ .

$f (x + h) = \frac{x^{2}}{3} + \frac{2 h x}{3} + \frac{h^{2}}{3} - 4$

Step 3.1.2.2

The final answer is $\frac{x^{2}}{3} + \frac{2 h x}{3} + \frac{h^{2}}{3} - 4$ .

$\frac{x^{2}}{3} + \frac{2 h x}{3} + \frac{h^{2}}{3} - 4$

Step 3.2

Reorder.

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Step 3.2.1

Move $\frac{x^{2}}{3}$ .

$\frac{2 h x}{3} + \frac{h^{2}}{3} + \frac{x^{2}}{3} - 4$

Step 3.2.2

Reorder $\frac{2 h x}{3}$ and $\frac{h^{2}}{3}$ .

$\frac{h^{2}}{3} + \frac{2 h x}{3} + \frac{x^{2}}{3} - 4$

Step 3.3

Find the components of the definition.

$f (x + h) = \frac{h^{2}}{3} + \frac{2 h x}{3} + \frac{x^{2}}{3} - 4$

$f (x) = \frac{x^{2}}{3} - 4$

$f (x + h) = \frac{h^{2}}{3} + \frac{2 h x}{3} + \frac{x^{2}}{3} - 4$

$f (x) = \frac{x^{2}}{3} - 4$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{h^{2}}{3} + \frac{2 h x}{3} + \frac{x^{2}}{3} - 4 - (\frac{x^{2}}{3} - 4)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{\frac{h^{2}}{3} + \frac{2 h x}{3} + \frac{x^{2}}{3} - 4 - \frac{x^{2}}{3} - - 4}{h}$

Step 5.1.2

Multiply $- 1$ by $- 4$ .

$\frac{\frac{h^{2}}{3} + \frac{2 h x}{3} + \frac{x^{2}}{3} - 4 - \frac{x^{2}}{3} + 4}{h}$

Step 5.1.3

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

$\frac{\frac{h^{2}}{3} + \frac{2 h x}{3} + 0 - 4 + 4}{h}$

Step 5.1.4

Add $\frac{h^{2}}{3}$ and $0$ .

$\frac{\frac{h^{2}}{3} + \frac{2 h x}{3} - 4 + 4}{h}$

Step 5.1.5

Add $- 4$ and $4$ .

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

Step 5.1.6

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

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

Step 5.1.7

Combine the numerators over the common denominator.

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

Step 5.1.8

Simplify the numerator.

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Step 5.1.8.1

Move $h$ .

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

Step 5.1.8.2

Reorder $h^{2}$ and $2 x h$ .

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

Step 5.1.9

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

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Step 5.1.9.1

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

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

Step 5.1.9.2

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

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

Step 5.1.9.3

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

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3

Cancel the common factor of $h$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 6