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 [(x_{1}, y_{1}) and (x_{2}, y_{2})] 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.
f_{1}(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.
f_{2}(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 marketclearing 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
