Linear functions are those whose graphs form straight lines. The formulas for linear functions can be expressed in several standard forms. The most common form, and the one used for this course, is the slope-intercept form.

The x and y values corresponding to the linear function are the points along the line. Graphing a line is simply a matter of substituting values for x into the function and plotting the resulting (x, y) ordered pairs. You need only plot two points to draw the line.

In the slope-intercept form, the parameter b represents the y-intercept of the line. The y-intercept is the point where the line crosses the vertical axis. To illustrate this point, consider the function f(x) = x + 10. The parameter b in this function is 10, which means the line crosses the vertical axis at the point (0, 10), because a point on the vertical axis always has an x-coordinate of 0.

In the slope-intercept form, the parameter m represents the slope of the line. The slope is a number that measures the steepness of the line. Slope can be positive, zero, or negative. (The slope may also be undefined, but that quality of slope will not be discussed here since it is not a function.) When m is positive, the slope is positive and results in a line moving up to the right. When m is equal to zero, the slope is zero and results in a horizontal line. When m is negative, the slope is negative and results in a line moving down to the right.

To illustrate this point, consider the linear function
f(x) = x. This function has been simplified from its original slope-intercept form, f(x) = 1x + 0. In this function, the parameter m is 1; therefore, the slope is positive, as shown below.

Slope is important because it describes the influence of the independent variable on the dependent variable. A larger value of slope, either positive or negative, indicates that the independent variable has a greater influence on the dependent variable. A slope of zero indicates that the independent variable does not influence the dependent variable at all. For linear functions, the slope is the same at every point along the line. Selecting any two points will yield the same result for the calculation of slope.

Slope is a ratio of vertical change (y) to horizontal change (x), sometimes referred to as rise over run.

If you do not know the value of m, you can calculate the slope using the coordinates of two points on the line. For example, find the slope of the line passing through the points (5, 0) and (13, 16). You can find the slope using the graph by measuring how many units are required to move from the first point to the second point. To move from the first point to the second, you would move 16 units vertically (rise) and 8 units horizontally (run). Because slope is the ratio of the rise (y) to the run (x), the slope of this line is , or 2.

You can calculate slope without using a graph. Using two points [(x₁, y₁) and (x₂, y₂)] on the line, you can determine the slope with the following formula:

From the example above, use the formula and the coordinates (5, 0) and (13, 16) to calculate the slope of the line.

Slope has important implications for business. For example, the two graphs below represent the impact on number of calls coming into the customer service center of the credit card company as customers began to use the company's website. The independent variable in these graphs is the customer's website usage, and the dependent variable is the number of calls coming into the customer service center.

The company has long been interested in getting the customer service website up and running in hopes of routing call traffic directly to the web and saving money in their call center.

The graph on the left shows the impact on the number of calls for account balance inquiries as a result of website usage. You can see that the slope of the line in the graph on the left is steep, indicating that the call volume has dropped off significantly as a result of website usage. A customer can easily log on to check his or her account balance without needing to consult with a customer service representative.

The graph on the right, on the other hand, has a gradual slope. This is the line for billing inquiries, and the slope tells you that the drop in calls to the service center has not been significant as a result of increased website usage. When customers have billing questions, they still prefer to speak with a customer service representative to get answers.

1. The function f(x) = 10x – 5 crosses the vertical axis at what point? Is the slope of this line positive, negative, or zero?

Solution 1

2. What is the slope of the line that passes through the points (0, 10) and (–2, 0)? Is the slope of this line positive, negative, or zero?

Solution 2

3. A company manufactures CDs and DVDs. Because the materials used to produce both products are similar, the number of CDs that can be produced depends on the number of DVDs produced. This relationship is modeled by the following linear function.

f₁(x) = –2x + 30

In this function, f(x) = the number of CDs produced, and x is the number of DVDs produced. A new type of machinery is introduced that makes the production of DVDs faster and easier. The new relationship is modeled by the following function.

f₂(x) = –1.5x + 30

Using this information, answer the following questions:

a. Before the introduction of the new machinery, how many CDs could be produced if 10 DVDs were produced?

b. After the introduction of the machinery, how many CDs could be produced if 10 DVDs were produced?

c. Which function has a steeper slope?

Solution 3

4. The demand for car radios is modeled by the following linear function, where f(x) is price and x is quantity demanded.

f(x) = –0.66x + 100

The supply of car radios is modeled by another linear function, where g(x) is the price, and x is the quantity supplied:

g(x) = x

Using this information, answer the following questions:

a. The market clearing price is the point where
f(x) = g(x). What is the market clearing price of these two functions? What is the quantity of radios at the market clearing price?

b. A shortage occurs when the quantity demanded is greater than the quantity supplied. A surplus occurs when the quantity supplied is greater than the quantity demanded. When the price, x, is $55, is there a surplus or a shortage? How great is it?

c. With the introduction of car CD players, the demand for car radios has declined. The new function is h(x) = –x + 100. What is the new market-clearing price and quantity, where h(x) = g(x)?

d. Which demand curve is steeper, f(x) or h(x)?

e. With the new demand, is there a shortage or a surplus at $55, and how great is it?

Solution 4a

Solution 4b

Solution 4c

Solution 4d

Solution 4e

Ratios