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High School Algebra II: Help and Review26 chapters | 296 lessons

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Lesson Transcript

Instructor:
*Kimberly Osborn*

While it can seem daunting, the difference quotient is a great tool to find the slope of the secant line of a curve. This lesson will break down the difference quotient into manageable steps so that you can become an expert at using this occasionally tricky formula.

When you first heard the term 'difference quotient', you may have drawn a blank. After all, it involves so many elements, like functions, and secants, and graphs, and even worse, a crazy formula that brings it all together. However, let me assure you, once you finish this lesson, you will definitely be an expert at using the difference quotient. Let's start with the definition: The **difference quotient** is used to calculate the slope of the secant line between two points on the graph of a function, *f*.

Just to review, a function is a line or curve that has only one *y* value for every *x* value. It's like an input/output machine. For any number *x* that you plug into the function, you will get an output value for *f* (*x*).

In simple terms, the difference quotient helps us find the slope when we are working with a curve. In the case of a curve, we cannot use the traditional formula of:

which is why we must use the difference quotient formula.

In the formal definition of the difference quotient, you'll note that the slope we are calculating is for the secant line. A **secant line** is just any line that passes between two points on a curve. We label these two points as *x* and (*x* +*h*) on our x-axis. Because we are working with a function, these points are labeled as *f* (*x*) and *f* (*x* + *h*) on our y-axis, respectively.

Now that we understand the definition of the difference quotient, let's explore the formula.

The first step necessary to finding the difference quotient is to find our *f* (*x* + *h*). When working with a function, all you have to do is plug (*x* + *h*) into your function wherever you see an *x*.

Let's look at the function: *f* (*x*) = 2*x* + 6.

To find our *f* (*x* + *h*), we need to plug in (*x* + *h*) into the function wherever we see an *x*.

Once we have our (*x* + *h*) plugged into the function, we must simplify our expression. In this case, we multiplied everything inside the parentheses by two to get:

F(x+h) = 2x + 2h + 6

Let's look at a more difficult function: *f* (*x*) = 3*x*^2 + 4.

Again, we must plug (*x* + *h*) into the function wherever we see an *x*.

Now, to simplify the expression, we must use the FOIL method to expand our (*x* + *h*)^2. Remember, FOIL stands for:

- Multiply the
**first**numbers: 3(x+h)^2 - Multiply the
**outer**numbers: 3(x+h) (x+h) - Multiply the
**inner**numbers: 3 (x^2 +xh+xh+h^2) - Multiply the
**last**numbers: 3 (x^2+2xh+h^2)

We then plug our foiled expression back into the function to get:

To finish simplifying the expression, we multiply everything inside the parentheses by three.

Not too difficult so far, right?

Now that we understand how to find *f* (*x* + *h*), we can plug our values into the difference quotient formula and simplify from there.

Let's use our earlier example of *f* (*x*) = 3*x*^2 + 4.

We can plug in the expression we found for *f* (*x* + *h*) and our expression for *f* (*x*) to get a difference quotient of:

It is extremely important to keep each segment of the difference quotient inside of it's own set of parentheses. This means that *f* (*x* + *h*) should be inside of it's own set of parentheses and *f* (*x*) should be inside of it's own set of parentheses.

This step is where some students make mistakes when working with the difference quotient. It is important to take note of the subtraction sign between *f* (*x* + *h*) and *f* (*x*). This subtraction sign tells us to change the sign in everything inside of the parentheses to the right of it. This allows us to get rid of our parentheses.

Once we have the expression written out with the parentheses removed, we can begin simplifying our terms.

The first thing to take note of is if there are any terms that are the same. In this case, we have a 3*x*^2 and a -3*x*^2. Because they are opposite signs, they cancel each other. We also have a positive four and negative four that cancel each other out.

After cleaning up the expression, we are left with:

Looking at our numerator, we see that both numbers share an *h*. This means that we can factor out an *h* from our numerator.

Now, we can clearly see that both our numerator and denominator are factors of *h*. Our rules of fractions tell us that we can then cancel these out to give us our simplified difference quotient.

We did it! When you approach difference quotients as a series of steps, the problems become a lot more manageable!

Now that we have the tools to find the difference quotient, let's solve this function: *f* (*x*) = 2*x*^2 + 4*x* - 3.

Remember, you should first find your *f*(*x* + *h*) by plugging in (*x* + *h*) wherever you see an *x*. Then plug your *f* (*x* + *h*) and *f* (*x*) into the difference quotient. Don't forget your parentheses! Finish by changing all of the signs after the minus between *f* (*x* + *h*) and *f* (*x*) and look for things you can cancel to simplify the expression. The answer is:

4x+2h+4

In this lesson, you became an expert at using the **difference quotient** to find the slope of the **secant line** for a given function. In particular, you learned how to break apart each piece of the difference quotient to make it much easier to solve.

First, plug (*x* + *h*) into your function wherever you see an *x*. Once you find *f* (*x* + *h*), you can plug your values into the difference quotient formula and simplify from there. In the third step, you use the subtraction sign to eliminate the parentheses and simplify the difference quotient.

**Difference quotient** - used to calculate the slope of the secant line between two points on the graph of a function, *f*

**Secant line** - any line that passes between two points on a curve

**FOIL** - a mnemonic to help remember the order for multiplying two binomials: first, outer, inner, last

Using this lesson to gather information about the difference quotient could prepare you to:

- Locate a secant line on the graph of a function
- Record the difference quotient formula
- Find the difference quotient for a secant line

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High School Algebra II: Help and Review26 chapters | 296 lessons

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