Lesson Objectives
• Learn how to Find the Difference Quotient

## How to Find the Difference Quotient

In this lesson, we will talk about the difference quotient. Let's revisit our formula for the slope of a line. Let’s suppose we have two points on a given line of P(x1,y1) and Q(x2,y2): We can use our slope formula to calculate the slope of the line that passes through these given points: $$m=\frac{y_2 - y_1}{x_2 - x_1}, x_2 - x_1 ≠ 0$$ Now let’s change this notation up and use our function notation. So our leftmost point will now just be: $$P(x,f(x))$$ And let’s say that the horizontal distance between the two points, will be labeled as h. This means our rightmost point could be written as: $$Q((x + h), f(x + h))$$ Now using this new notation, we can show the slope as: $$m=\frac{f(x + h) - f(x)}{x + h - x}, h ≠ 0$$ We can further simplify this as: $$m=\frac{f(x + h) - f(x)}{h}, h ≠ 0$$ This is still the difference in y-values over the difference in x-values. We are just using function notation. This expression is known as the difference quotient.

### The Difference Quotient:

$$m=\frac{f(x + h) - f(x)}{h}, h ≠ 0$$ Example #1: Find the Difference Quotient. $$f(x)=3x^2 - 4$$ Our goal is to find [f(x + h) - f(x)] / [h]. To do this, let's start by finding f(x + h).
Step 1) Replace x with x + h: $$f(x + h)=3(x + h)^2 - 4$$ $$f(x + h)=3(x^2 + 2xh + h^2) - 4$$ $$f(x + h)=3x^2 + 6xh + 3h^2 - 4$$ Step 2) We can now plug into our formula: $$\frac{f(x + h) - f(x)}{h}, h ≠ 0$$ $$\frac{3x^2 + 6xh + 3h^2 - 4 - (3x^2 - 4)}{h}$$ $$\frac{3x^2 + 6xh + 3h^2 - 4 - 3x^2 + 4}{h}$$ $$\frac{6xh + 3h^2}{h}$$ $$\require{cancel}\frac{\cancel{h}(6x + 3h)}{\cancel{h}}$$ $$6x + 3h$$

#### Skills Check:

Example #1

Find the difference quotient. $$f(x)=5x^2 - 4x + 3$$

A
$$3x + 7h - 3$$
B
$$10x + 5h - 4$$
C
$$8x + 9h + 7$$
D
$$-7x - 5h + 2$$
E
$$17x + 9h + 2$$

Example #2

Find the difference quotient. $$f(x)=9x - 7$$

A
$$7x - 3h$$
B
$$-3$$
C
$$9x - 4h$$
D
$$9$$
E
$$3$$

Example #3

Find the difference quotient. $$f(x)=\frac{1}{x - 19}$$

A
$$\sqrt{x - 19}+ 19h$$
B
$$x - 19xh - 3$$
C
$$\frac{15x}{(x + h - 19)(x - 19)}$$
D
$$-\frac{1}{(x + h - 19)(x - 19)}$$
E
$$-\frac{19}{(x - h - 19)(x - 19)}$$

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