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): difference quotient 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: $$m=\frac{f(x + h) - f(x)}{h}, h ≠ 0$$ difference quotient, shows h 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. $$f(x)=5x^2 - 4x + 3$$

Please choose the best answer.

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. $$f(x)=9x - 7$$

Please choose the best answer.

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

Example #3

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

Please choose the best answer.

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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