Precalculus Examples

$y = 100 - 15 x$

Step 1

Write $y = 100 - 15 x$ as a function.

$f (x) = 100 - 15 x$

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.

$f (x + h) = 100 - 15 (x + h)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = 100 - 15 x - 15 h$

Step 3.1.2.2

The final answer is $100 - 15 x - 15 h$ .

$100 - 15 x - 15 h$

Step 3.2

Reorder.

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Step 3.2.1

Move $100$ .

$- 15 x - 15 h + 100$

Step 3.2.2

Reorder $- 15 x$ and $- 15 h$ .

$- 15 h - 15 x + 100$

Step 3.3

Find the components of the definition.

$f (x + h) = - 15 h - 15 x + 100$

$f (x) = 100 - 15 x$

$f (x + h) = - 15 h - 15 x + 100$

$f (x) = 100 - 15 x$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 15 h - 15 x + 100 - (100 - 15 x)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $5$ out of $100 - 15 x$ .

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Step 5.1.1.1

Factor $5$ out of $100$ .

$\frac{- 15 h - 15 x + 100 - (5 (20) - 15 x)}{h}$

Step 5.1.1.2

Factor $5$ out of $- 15 x$ .

$\frac{- 15 h - 15 x + 100 - (5 (20) + 5 (- 3 x))}{h}$

Step 5.1.1.3

Factor $5$ out of $5 (20) + 5 (- 3 x)$ .

$\frac{- 15 h - 15 x + 100 - (5 (20 - 3 x))}{h}$

$\frac{- 15 h - 15 x + 100 - 1 \cdot 5 (20 - 3 x)}{h}$

Step 5.1.2

Multiply $- 1$ by $5$ .

$\frac{- 15 h - 15 x + 100 - 5 (20 - 3 x)}{h}$

Step 5.1.3

Factor $5$ out of $- 15 h - 15 x + 100 - 5 (20 - 3 x)$ .

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Step 5.1.3.1

Factor $5$ out of $- 15 h$ .

$\frac{5 (- 3 h) - 15 x + 100 - 5 (20 - 3 x)}{h}$

Step 5.1.3.2

Factor $5$ out of $- 15 x$ .

$\frac{5 (- 3 h) + 5 (- 3 x) + 100 - 5 (20 - 3 x)}{h}$

Step 5.1.3.3

Factor $5$ out of $100$ .

$\frac{5 (- 3 h) + 5 (- 3 x) + 5 (20) - 5 (20 - 3 x)}{h}$

Step 5.1.3.4

Factor $5$ out of $- 5 (20 - 3 x)$ .

$\frac{5 (- 3 h) + 5 (- 3 x) + 5 (20) + 5 (- (20 - 3 x))}{h}$

Step 5.1.3.5

Factor $5$ out of $5 (- 3 h) + 5 (- 3 x)$ .

$\frac{5 (- 3 h - 3 x) + 5 (20) + 5 (- (20 - 3 x))}{h}$

Step 5.1.3.6

Factor $5$ out of $5 (- 3 h - 3 x) + 5 (20)$ .

$\frac{5 (- 3 h - 3 x + 20) + 5 (- (20 - 3 x))}{h}$

Step 5.1.3.7

Factor $5$ out of $5 (- 3 h - 3 x + 20) + 5 (- (20 - 3 x))$ .

$\frac{5 (- 3 h - 3 x + 20 - (20 - 3 x))}{h}$

Step 5.1.4

Apply the distributive property.

$\frac{5 (- 3 h - 3 x + 20 - 1 \cdot 20 - (- 3 x))}{h}$

Step 5.1.5

Multiply $- 1$ by $20$ .

$\frac{5 (- 3 h - 3 x + 20 - 20 - (- 3 x))}{h}$

Step 5.1.6

Multiply $- 3$ by $- 1$ .

$\frac{5 (- 3 h - 3 x + 20 - 20 + 3 x)}{h}$

Step 5.1.7

Add $- 3 x$ and $3 x$ .

$\frac{5 (- 3 h + 0 + 20 - 20)}{h}$

Step 5.1.8

Add $- 3 h$ and $0$ .

$\frac{5 (- 3 h + 20 - 20)}{h}$

Step 5.1.9

Subtract $20$ from $20$ .

$\frac{5 (- 3 h + 0)}{h}$

Step 5.1.10

Add $- 3 h$ and $0$ .

$\frac{5 \cdot - 3 h}{h}$

Step 5.1.11

Multiply $5$ by $- 3$ .

$\frac{- 15 h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{- 15 h}{h}$

Step 5.2.2

Divide $- 15$ by $1$ .

$- 15$

Step 6