Questions on Linear Functions


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This page contains sample problems on linear functions. They are for Self-assessment and Review.

Each problem (or group of problems) has an "answer button" ANSWER which you can click to look at an answer. Some solutions have a "further explanation button" Explain which you can click to see a more complete, detailed solution.

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To gain the most benefit from these problems,

Work the problems on your own. Write down your solutions BEFORE looking at any answers.

Use a graphing calculator as appropriate. A graphing calculator can be used to verify that your answers "make sense" or "look right".

If you have difficulties with this material, please contact your instructor. (See Getting Help in Stage 1.) It will be very difficult to succeed in Calculus without being able to solve and manipulate linear equations.


1. Find the slopes and the x- and y-intercepts of the following lines.

  1. y + 3 = -2 (x - 5)

  2. y = 1.2 x - 7

  3. 3 x - 5 y = 20

  4. y - c = 2 x + c/2
ANSWER

2. In economics the demand function relates the price per unit of an item to the number of units that consumers will buy at that price. The demand, q, is considered to be the independent variable, while the price, p, is considered to be the dependent variable.

Suppose that in a certain market, the demand function for widgets is a linear function

p = -0.75q + 54,

where p is the price in dollars and q is the number of units (hundreds of widgets in this case).

  1. What is the slope of this function? Explain the meaning of the sign of the slope in practical terms.

  2. Find the p- and q- intercepts for this function. What is the significance of these intercepts in the context of the problem?

ANSWER

3. State whether the following pairs of lines are parallel, perpendicular or neither:

  1. y = (3/2)x -7 and 3x - 2y = 4

  2. 5x - 3y = 12 and 3x + 5y = 10

  3. x - y = 10 and x + y = -1

  4. x - 2y = 1 and 2x - y = 5

  5. x - 3y = 5 and -2x + 6y = 8

  6. 3x + 7y = 9 and -6x + 14y = 21

  7. y = (2/5)x + 2 and 5x - 2y = -4

  8. x = 10 and y - 10 = 0

ANSWER

4. Find the equation of each of the following lines:

  1. The line with slope -1/2 and passing through the point (0, 3).

  2. The line with slope -2/3 and containing the point (6, -1).

  3. The line passing through the points (7, -1) and (4, 5).

  4. The line with slope 6 and passing through the graph of f(x) = x2 where x = 3.

  5. The line passing through (4,0) and the graph of f(x) = x2/3 where x = -8.

  6. The line perpendicular to 3x + y = 17 and passing through ( 15, 2.5).
ANSWER

5. A small college has 2546 students in 1994 and 2702 students in 1996. Assume that the enrollment follows a linear growth pattern. Let t = 0 correspond to 1990 and let y(t) represent the enrollment in year t.

  1. Assume that y(t) is linear. Using the data given, find the slope of y(t).

  2. What does the slope of y(t) signify in terms of enrollment growth?

  3. Find an equation for y(t) and use it predict the enrollment of the college in 1999.

ANSWER

6. Find the point(s) of intersection of each of the following pairs of lines.

  1. 2 x - y = 10 and x + y = -1

  2. y = 2 x + 5 and y - 1 = 2 (x -3)

  3. y = (2/3) x + 5 and 2 x - 3 y = -15

  4. 3 x + 3 y = 180 and 3.6 x - 3.6 y = 180
ANSWER


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