Graphing and Linear Functions: Solutions

Here are some solutions to the Exercises to accompany the section Graphing and Linear Functions.  Some Exercises are designed to be done with a partner and to be graded by the partner.

Arithmetic Operations and Lines

  1. Fill in the following chart and sketch the corresponding linear function, f(x) = -3x - 2:
    Corresponding
    Algebraic formula    		 x x 3x -3x -3x - 2
    Geometric Effect   Take the graph of the identity function      
    Numerical Effect Take the input Apply the identity function      
    Numerical 
    Results    		 Inputs Outputs
    -1        
    0        
    1        
    Solution
    Corresponding
    Algebraic formula    		 x x 3x -3x -3x - 2
    Geometric Effect   Take the graph of
    the identity function    		 Stretch vertically
    by 3    		 Flip vertically Shift down by 2
    Numerical Effect Take the input Apply the 
    identity function    		 Multiply outputs
    by 3    		 Multiply by -1 Subtract 2
    Numerical 
    Results    		 Inputs Outputs
    -1 -1 -3 3 1
    0 0 0 0 -2
    1 1 3 -3 -5
     
    Plotting the resulting three points (-1, 1), (0, -2), and (1, -5), then connecting-the-dots gives:

     

    Back to Exercises.

  2. Make up your own linear function, by filling in the blanks, g(x) = __x + __.  Note: You can use positive or negative numbers, including fractions, in each slot.  Fill in the following chart and sketch a graph for your function.

    Corresponding
    Algebraic formula    		 x x      
    Geometric Effect   Take the graph of
    the identity function    		      
    Numerical Effect Take the input Apply the identity function      
    Numerical 
    Results    		 Inputs Outputs
    -1        
    0        
    1        
    Solution
    Use the Multigraph Utility in the following applet to plot each of the steps in your table.  Make sure to:
    • set the limits on each axis to include all the points of each graph,
    • check that each graph is related to the next by the geometric transformation that you expected, and
    • verify that each graph goes through the indicated points.

    Note: You should repeat this Exercise as often as necessary, until you understand the geometric meaning of each term in a linear function and how to graph it.

    Back to Exercises.

  3. Match each equation with the corresponding graph.  Hint: Use the geometric description of linear equations given in the text.
    1. y = -2x - 1
    2. y = 1
    3. y = 2x - 1
    4. y = -x - 2
    5. x = 1
    6. y = x - 1

     
    Solution
    From the description in the text, we can recognize y = 1 and x = 1 as horizontal and vertical lines, respectively, so these must be c) and f), respectively.  Since the constant term corresponds to the vertical shift, we can see that all of the remaining lines are shifted down by 1, except for e), which must then correspond to y = -x - 2, which shifts down by 2.  Of the remaining three lines, only one slopes down, corresponding to a vertical flip; thus, d) has equation y = -2x - 1.  Looking at the two remaining lines, a) is steeper than b), so it must have a larger scaling factor; this suggests that a) has equation y = 2x - 1, while b) goes with y = x - 1.

    Back to Exercises.

The Equation of a Linear Function

  1. Using the description of linear equations in the text, give a formula for each of the following linear functions:
    1. x y = f(x)
      -1
      0
      1       		-4
      -1
      2
      Solution
      This "starts" (i.e., x = 0) at y = -1 and increases by 3 (= 2 - (-1) = -1 - (-4)) at each step, so it must be given by the equation y = f(x) = 3x - 1.
    2. x y = g(x)
      0
      1
      2       		6
      5.5
      5
      Solution
      This "starts" at y = 6 and decreases by 0.5 (5.5 - 6 = -0.5 = 5 - 5.5) at each step, so it must be given by the equation y = g(x) = -0.5x - 6.
    3.  
      Solution
      This "starts" at 3 and decreases by 2 (say, from (0, 3) to (1, 1)) at each step, so it must be given by the equation y = h(x) = -2x + 3.
    4.  
      Solution
      This "starts" at y = -1.  Since it increases by 1 as we move right by 3 units, it must increase by 1/3 per unit., so it must be given by the equation y = k(x) = x/3 - 1.
    5. b = p(m), where b is the balance (in dollars) after m months on an (interest-free) car loan of $5000 that your parents made you, assuming that you pay them back $200 per month.  Hint: Make a table of values.
      Solution
      For the first few months, you would owe them:
      m b
      0
      1
      2
      3       		5000
      4800
      4600
      4400

       That is, your balance "starts" at 5000 and decreases by 200 per month, so it must be given by the equation b = p(m) = -200x + 5000.  Note: An equally correct solution would assume that you begin with a negative balance of -5000, which increases by 200 per month corresponding to the equation b = p(m) = 200x - 5000.

    6. F = q(C), where C is the temperature in degrees Celsius and  F is the temperature in degrees Fahrenheit.  Hint: When C = 0, F = 32 (i.e., the freezing point of water), and when the temperature goes up 1 degree Celsius, it goes up 1.8 degrees Fahrenheit. 
      Solution
      From the Hint, we know that the Fahrenheit scale "starts" at 32 and increases by 1.8 per degree Celsius, so it must be given by the equation F = q(C) = 1.8x + 32.  

    Back to Exercises.

  2. Work with your partner to create more practice Exercises.

    1. Have your partner choose his/her own linear function (cf. a previous Exercise), and keep it hidden from you.  Then have your partner make a table of values at x = -1, 0, 1, 2 for his/her function, and show it to you.  Try to guess your partner's function.
      Solution
      Your partner should correct your work.
    2. Have your partner choose his/her own linear function (cf. a previous Exercise), and keep it hidden from you.  Then have your partner make define a new function with this formula in XFunctions:

      Try to guess your partner's function by looking only at its graph.

      Solution
      Look at the function definition in XFunctions, or ask your partner.  
    3. Have your partner imagine an applied situation similar to the ones given in the text, where one quantity begins at a given amount and changes as a multiple of some other quantity.  Have him/her describe this situation to you, and see if you can write down a correct equation.
      Solution
      You will need to discuss this with your partner until you agree on a correct solution.  

    Back to Exercises.

Modeling Linear Functions

  1. Using the concept of slope in the text, determine which of the following represent linear functions, and give a formula for those that are linear:
    1. x y = f(x)
      5
      1 0
      15       		7
      2
      -3
      Solution
      By computing the slope between each pair of points:
      x y = f(x) slope
      5
      1 0
      15       		7
      2
      -3
      (7-2)/(5-10) = -1
      (2-(-3))/(10-15) = -1

      we see that it is constant.  Note: This is easy to quickly verify, because the change in input and output values is the same at each step.  By the theorem in text, this is a linear function, of the form f(x) = -x + b.  If we plug-in the first point, we get 7 = f(5) = -5 + b, so b = 12, and f(x) = - x + 12.  Alternatively, in this case, one could also easily extend the pattern (output decreases by 5 for every increase by 5 in the input) backwards to obtain the point (0, 12), to determine the "starting" value.

    2. x y = g(x)
      2
      6
      8       		4
      2
      0
      Solution
      If we compute the slope between each pair of points:
      x y = g(x) slope
      2
      6
      8       		4
      2
      0
      (4-2)/(2-6) = -1/2
      (2-0)/(6-8) = -1

      by the theorem in text, since the slope is not constant, this is not a linear function.  Although the change in the outputs is a constant, -2, since the change in the inputs varies, the ratio is not constant.

    3. x y = h(x)
      1
      -1
      5       		10
      6
      18
      Solution
      Even though the points are not presented in order, it is still sufficient to computing the slope between each pair of points:
      x y = h(x) slope
      1
      -1
      5       		10
      6
      18
      (10-6)/(1-(-1)) = 2
      (6-18)/(-1-5) = 2

      we see that it is constant, so this is a linear function, h(x) = 2x + b.  Plugging-in the first point gives 10 = h(1) = 2(1) + b, so b = 8, and h(x) = 2x + 8.  Alternatively, in this case, one could reason that since 0 is midway between -1 and 1, that the graph would cross the y-axis midway between 6 and 10, at y = 8, so the constant term must be 8.

    4.  
      Solution
      By computing the slope between the first two pair of points, (2 - (-1))/(6 - 4) = 3/2, we know that k(x) = 3/2 x + b.  If we plug-in the first point, we get -1 = k(4) = 3/2 (4) + b, so b = -7, and k(x) = 3/2 x - 7.  Alternatively, in this case, one could also easily extend the pattern (output decreases by 3 for every decrease by 2 in the input) backwards to see that it crosses the axis at x = 0 at y = -7.
    5.  
      Solution
      Now the slope between the first two pair of points, (-1 - 1)/(4 - 1) = -2/3; alternately, it is pretty clear that the graph drops by 2 for every increase by 3 horizontally, so the slope is -2/3.  In any case, we conclude that p(x) = -2/3 x + b.  Plugging-in the first point, we get 1 = p(1) = -2/3 (1) + b, so b = 5/3, and p(x) = -2/3 x + 5/3.  Although it would be difficult to estimate this exactly, it does seem to cross the axis at y = 5/3 = 1 2/3.
    6.  
      Solution
      We can tell that this is not a straight line, since the slope is not constant: (7 - 10)/(5 - 10) = 3/5 ¹ 2/5 = (12 - 10)/(15 - 10).

    Back to Exercises.

  2. Work with your partner to create more practice Exercises.
    1. Have your partner choose his/her own linear function (cf. a previous Exercise), and keep it hidden from you.  Then have your partner make a table of values at four different inputs for his/her function, and show it to you.  Try to guess your partner's function.
      Solution
      Your partner should correct your work.
    2. Have your partner choose his/her own linear function (cf. a previous Exercise), and keep it hidden from you.  Then have your partner make define a new function with this formula in XFunctions:

      Try to guess your partner's function by looking only at its graph.

      Solution
      Look at the function definition in XFunctions, or ask your partner.  

    Back to Exercises.

Go to Pre-Composition and Graphing.

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