The Geometric Interpretation of Difference Quotient

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  • 0:01 A Function
  • 0:29 Difference Quotient
  • 1:26 Geometric Interpretation
  • 2:41 Example
  • 3:39 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will understand how the difference quotient looks from a geometric standpoint. You will also see how the difference quotient is applied to a function.

A Function

In this video lesson, we begin with a function. Let's look at the function f(x) = x - 3. If we graph it out, what do we get? From your previous studies on functions, you realize that you will get a straight line.

Graph of f(x) = x - 3
difference quotient

Since we are studying pre-calculus right now, we are on our way to working more extensively with these functions. This is just what we are going to do in this lesson. We are going to manipulate our function using the difference quotient.

Difference Quotient

The difference quotient takes two points from our function, points (x, f(x)) and (x + h, f(x + h)), and plugs them into this formula:

difference quotient

Because we have moved beyond algebra now, we leave the x and h variables as they are. When we plug these points into our formula, we keep them as they are. We won't be replacing them with numbers. We will expect to get an answer that most likely will have these variables in it. Sometimes, our answer will work out so that we get a nice number, but most times you will get an answer with variables in it. So, as long as you keep your like terms together, you will be okay.

Geometric Interpretation

This difference quotient formula might look complicated, but it really isn't. Let me show you what it means from a geometric standpoint. We can plot two random points on our graph. The first point will be the point (x, f(x)). The next point will be the point (x + h, f(x + h)).

Now, if you draw a line through these two points, you will get what is called a secant line. The formula for the difference quotient then gives you the slope of this secant line. Remember that slope is rise over run. So the numerator part of the difference quotient gives you the rise while the denominator gives you the run. The h here stands for how far your run is and not the height.

Difference quotient
difference quotient

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