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Name ________________________________________ Date ___________________ Class __________________
Unit 4 Lesson 4
Challenge
When two variables x and y are related and (x1, y1) and (x2, y2) satisfy the
relationship, you can write the quotient as shown at right. The quotient
is called a difference quotient.
The difference quotient can have different meanings in different situations.
In Exercises 1–5, use the grid at the right.
1. Graph and connect A(−2, −3), B(1, 3), and C(3, 7).
2. Calculate the slope of
and of
. __________________________
3. On the same grid, graph D(2, 6). Connect A, B and D.
4. Calculate the slope of
and of
. __________________________
5. Explain how to use a difference quotient to determine whether three
points are collinear, that is, whether three points lie on the same line.
_________________________________________________________________________________________
Another application of the difference quotient is the measurement of the grade, or steepness,
of a road. A grade of 3% means that the road surface rises 3 feet for every horizontal run of
100 feet. A roadway that slopes down has a negative grade.
6. If the vertical rise of a highway is 150 feet when the
horizontal run is 5000 feet, what is the grade of the road?
__________________________
7. If the grade of a road is 5%, what is the vertical rise
of the road when the horizontal run is 6000 feet?
__________________________
8. On a certain stretch of road, the point 420 feet east of
point A is 24 feet lower than point A. Point B, 600 feet
east of A, is 36 feet lower than point A.
a. Find the grade of the road surface between points A and B. __________________________
b. Would a motorist driving along a stretch of road from point B
to point A report the same grade for the road as a motorist
driving from A to B would report? Explain your response.
____________________________________________________________________________________
In economics, the additional cost to produce one more unit of an item is
called the marginal cost of the item. Marginal cost is found by computing
the difference quotient, or difference in
9. The cost that a manufacturer pays for producing 5 pairs of shoes is $113.
The cost to produce 12 pairs of shoes is $127. Use a difference quotient to find
the marginal cost.
_______________________________________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Algebra 1
Name ________________________________________ Date ___________________ Class __________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Algebra 1
Name ________________________________________ Date ___________________ Class __________________
Challenge
Exploring the Meaning of a Difference Quotient
When two variables x and y are related and (x1, y1) and (x2, y2) satisfy the
relationship, you can write the quotient as shown at right. The quotient
is called a difference quotient.
The difference quotient can have different meanings in different situations.
In Exercises 1–5, use the grid at the right.
1. Graph and connect A(−2, −3), B(1, 3), and C(3, 7).
2. Calculate the slope of
Slope of
and of
. Slope of
=
=2
3. On the same grid, graph D(2, 6). Connect A, B and D.
4. Calculate the slope of
and of
. Slope of
Slope of
= 2;
=3
5. Explain how to use a difference quotient to determine whether three
points are collinear, that is, whether three points lie on the same line.
Points A, B, and C lie on a line if the slopes of
and
are equal.
Another application of the difference quotient is the measurement of the grade, or steepness,
of a road. A grade of 3% means that the road surface rises 3 feet for every horizontal run of
100 feet. A roadway that slopes down has a negative grade.
6. If the vertical rise of a highway is 150 feet when the
3%
horizontal run is 5000 feet, what is the grade of the road?
7. If the grade of a road is 5%, what is the vertical rise
300 feet
of the road when the horizontal run is 6000 feet?
8. On a certain stretch of road, the point 420 feet east of
point A is 24 feet lower than point A. Point B, 600 feet
east of A, is 36 feet lower than point A.
6%
a. Find the grade of the road surface between points A and B.
b. Would a motorist driving along a stretch of road from point B
to point A report the same grade for the road as a motorist
driving from A to B would report? Explain your response.
No; the uphill grade is positive and the downhill grade is negative.
In economics, the additional cost to produce one more unit of an item is
called the marginal cost of the item. Marginal cost is found by computing
the difference quotient, or difference in
9. The cost that a manufacturer pays for producing 5 pairs of shoes is $113.
The cost to produce 12 pairs of shoes is $127. Use a difference quotient to find
$2
the marginal cost.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Holt McDougal Algebra 1