Find the Difference Quotient f(x)=-9x+3

$f (x) = - 9 x + 3$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

$f (x + h) = - 9 (x + h) + 3$

Simplify the result.

Apply the distributive property.

$f (x + h) = - 9 x - 9 h + 3$

The final answer is $- 9 x - 9 h + 3$ .

$- 9 x - 9 h + 3$

Reorder $- 9 x$ and $- 9 h$ .

$- 9 h - 9 x + 3$

Find the components of the definition.

$f (x + h) = - 9 h - 9 x + 3$

$f (x) = - 9 x + 3$

$f (x + h) = - 9 h - 9 x + 3$

$f (x) = - 9 x + 3$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 9 h - 9 x + 3 - (- 9 x + 3)}{h}$

Step 4

Simplify.

Simplify the numerator.

Factor $3$ out of $- 9 x + 3$ .

Factor $3$ out of $- 9 x$ .

$\frac{- 9 h - 9 x + 3 - (3 (- 3 x) + 3)}{h}$

Factor $3$ out of $3$ .

$\frac{- 9 h - 9 x + 3 - (3 (- 3 x) + 3 (1))}{h}$

Factor $3$ out of $3 (- 3 x) + 3 (1)$ .

$\frac{- 9 h - 9 x + 3 - (3 (- 3 x + 1))}{h}$

$\frac{- 9 h - 9 x + 3 - 1 \cdot 3 (- 3 x + 1)}{h}$

Multiply $- 1$ by $3$ .

$\frac{- 9 h - 9 x + 3 - 3 (- 3 x + 1)}{h}$

Factor $3$ out of $- 9 h - 9 x + 3 - 3 (- 3 x + 1)$ .

Factor $3$ out of $- 9 h$ .

$\frac{3 (- 3 h) - 9 x + 3 - 3 (- 3 x + 1)}{h}$

Factor $3$ out of $- 9 x$ .

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

Factor $3$ out of $3$ .

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

Factor $3$ out of $- 3 (- 3 x + 1)$ .

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

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

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

Factor $3$ out of $3 (- 3 h - 3 x) + 3 (1)$ .

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

Factor $3$ out of $3 (- 3 h - 3 x + 1) + 3 (- (- 3 x + 1))$ .

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

Apply the distributive property.

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

Multiply $- 3$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

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

$\frac{3 (- 3 h + 0 + 1 - 1)}{h}$

Add $- 3 h$ and $0$ .

$\frac{3 (- 3 h + 1 - 1)}{h}$

Subtract $1$ from $1$ .

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

Add $- 3 h$ and $0$ .

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

Multiply $3$ by $- 3$ .

$\frac{- 9 h}{h}$

Divide $- 9$ by $1$ .

$- 9$

Step 5

Find the Difference Quotient f(x)=-9x+3

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