Precalculus Examples

$f (x) = a x^{2} + c x + d$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$f (x + h) = a {(x + h)}^{2} + c (x + h) + d$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

$f (x + h) = a ((x + h) (x + h)) + c (x + h) + d$

Step 2.1.2.1.2

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

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = a (x (x + h) + h (x + h)) + c (x + h) + d$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = a (x \cdot x + x h + h (x + h)) + c (x + h) + d$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = a (x \cdot x + x h + h x + h \cdot h) + c (x + h) + d$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = a (x^{2} + x h + h x + h \cdot h) + c (x + h) + d$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = a (x^{2} + x h + h x + h^{2}) + c (x + h) + d$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = a (x^{2} + h x + h x + h^{2}) + c (x + h) + d$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = a (x^{2} + 2 h x + h^{2}) + c (x + h) + d$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = a x^{2} + a (2 h x) + a h^{2} + c (x + h) + d$

Step 2.1.2.1.5

Rewrite using the commutative property of multiplication.

$f (x + h) = a x^{2} + 2 a h x + a h^{2} + c (x + h) + d$

Step 2.1.2.1.6

Apply the distributive property.

$f (x + h) = a x^{2} + 2 a h x + a h^{2} + c x + c h + d$

Step 2.1.2.2

The final answer is $a x^{2} + 2 a h x + a h^{2} + c x + c h + d$ .

$a x^{2} + 2 a h x + a h^{2} + c x + c h + d$

Step 2.2

Reorder.

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Step 2.2.1

Move $c x$ .

$a x^{2} + 2 a h x + a h^{2} + c h + c x + d$

Step 2.2.2

Move $a x^{2}$ .

$2 a h x + a h^{2} + a x^{2} + c h + c x + d$

Step 2.2.3

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

$a h^{2} + 2 a h x + a x^{2} + c h + c x + d$

Step 2.3

Find the components of the definition.

$f (x + h) = a h^{2} + 2 a h x + a x^{2} + c h + c x + d$

$f (x) = a x^{2} + c x + d$

$f (x + h) = a h^{2} + 2 a h x + a x^{2} + c h + c x + d$

$f (x) = a x^{2} + c x + d$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{a h^{2} + 2 a h x + a x^{2} + c h + c x + d - (a x^{2} + c x + d)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{a h^{2} + 2 a h x + a x^{2} + c h + c x + d - (a x^{2}) - (c x) - d}{h}$

Step 4.1.2

Remove parentheses.

$\frac{a h^{2} + 2 a h x + a x^{2} + c h + c x + d - a x^{2} - c x - d}{h}$

Step 4.1.3

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

$\frac{a h^{2} + 2 a h x + c h + c x + d + 0 - c x - d}{h}$

Step 4.1.4

Add $a h^{2}$ and $0$ .

$\frac{2 a h x + c h + c x + d + a h^{2} - c x - d}{h}$

Step 4.1.5

Subtract $c x$ from $c x$ .

$\frac{2 a h x + c h + d + a h^{2} + 0 - d}{h}$

Step 4.1.6

Add $2 a h x$ and $0$ .

$\frac{c h + d + a h^{2} + 2 a h x - d}{h}$

Step 4.1.7

Subtract $d$ from $d$ .

$\frac{c h + a h^{2} + 2 a h x + 0}{h}$

Step 4.1.8

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

$\frac{c h + a h^{2} + 2 a h x}{h}$

Step 4.1.9

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

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Step 4.1.9.1

Factor $h$ out of $c h$ .

$\frac{h c + a h^{2} + 2 a h x}{h}$

Step 4.1.9.2

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

$\frac{h (c) + h (a (h)) + 2 a h x}{h}$

Step 4.1.9.3

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

$\frac{h (c) + h (a (h)) + h (2 a x)}{h}$

Step 4.1.9.4

Factor $h$ out of $h (c) + h (a (h))$ .

$\frac{h (c + a (h)) + h (2 a x)}{h}$

Step 4.1.9.5

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

$\frac{h (c + a (h) + 2 a x)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{h (c + a (h) + 2 a x)}{h}$

Step 4.2.1.2

Divide $c + a h + 2 a x$ by $1$ .

$c + a h + 2 a x$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $c$ .

$a h + 2 a x + c$

Step 4.2.2.2

Reorder $a h$ and $2 a x$ .

$2 a x + a h + c$

Step 5