Powers and Proportions: Solutions

Here are some solutions to the Exercises to accompany the section Powers and Proportions.  

Power Functions

  1. Use the general properties of power functions to match the formulas with their graphs below:
    1. x5
    2. x4
    1. x3
    2. x2
    1. 1/x3
    2. 1/x
    1. 1/x4
    2. 1/x6

    Graphs iv) and v) look like odd power graphs, so they match with either a) or c).  Since iv) is steeper and flatter than v), it must correspond to the higher exponent, namely, a), while v) corresponds to c).   

    Similarly, graphs i) and viii) look like even power graphs, so they match with either b) or d).  Since viii) is steeper and flatter than i), it must correspond to the higher exponent, namely, b), while i) corresponds to d).  

    The remaining graphs have vertical asymptotes, so correspond to negative (i.e., reciprocal) power graphs.  Since they are symmetric with respect to the vertical axis, graphs iii) and vii) look like even power graphs.  Therefore, they match with either g) or h).  Since vii) is steeper and flatter than iii), it must correspond to the g), while vii) corresponds to h). 

    In the same way, graphs ii) and vi) look like odd power graphs, and correspond to f) and e), respectively.. 

    Back to Exercises.

  2. Use our transformational graphing technique to graph the following exponential functions:
    1. f(x) = 1/(-x)3 - 2
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		 1/x3 1/(-x)3   1/(-x)3 - 2
      Geometric Effect Take the power
      function with exponent -3      		 Flip horizontally Shift down 2
      Numerical Effect   Negate Apply the power
      function with exponent -3      		 Subtract 2
      Numerical 
      Results      		 Inputs Outputs
      -¥ ¥ 0 -2
      -1 1 -1 -3
      0 0 undefined undefined
      1 -1 1 -1
      ¥ -¥ 0 -2
      The final graph has a horizontal asymptote at y = -2 and a vertical asymptote at x = 0.  Since it also goes through the points (1, -3) and (-1, -1), it must look like this:

      It is similar to the 1/x graph, but approaches the horizontal asymptote more quickly, due to the higher exponent.  You can see how this graph is "built" out of the graph of 1/x3 by selecting the Examples in the following applet:

    2. g(x) = -3/(x + 1)2 + 1
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		 1/x2 1/(x + 1)2   3/(x + 1)2 -3/(x + 1)2 -3/(x + 1)2 + 1
      Geometric Effect Take the power
      function with exponent -2      		 Shift left 1 unit Stretch vertically
      by a factor of 3      		 Flip vertically Shift up 1
      Numerical Effect   Subtract 1 Apply the power
      function with exponent -2      		 Multiply by 3 Negate Add 1
      Numerical 
      Results      		 Inputs Outputs
      -¥ -¥ 0 0 0 1
      -1 -2 1 3 -3 -2
      0 -1 undefined undefined undefined undefined
      1 0 1 3 -3 -2
      ¥ ¥ 0 0 0 1
      The final graph has a horizontal asymptote at y = 1 and a vertical asymptote at x = -1.  Since it also goes through the points (-2, -2) and (0, -2), it must look like this:

      Notice how the graph is symmetric with respect to the vertical asymptote, because of the even power.  You can see how this graph is "built" out of the graph of 1/x2 by selecting the Examples in the following applet:

    3. h(x) = 4(x - 2)5 - 3
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		 x5 (x - 2)5   4(x - 2)5 4(x - 2)5 - 3
      Geometric Effect Take the 5th 
      power function      		 Shift right
      2 units      		 Stretch vertically
      by a factor of 4      		 Shift down 3
      Numerical Effect   Add 2 Raise to the 
      5th power      		 Multiply by 4 Subtract 3
      Numerical 
      Results      		 Inputs Outputs
      -1 1 -1 -4 -7
      0 2 0 0 -3
      1 3 1 4 1
      This graph is like a cube graph, but steeper and flatter:
      You can see how this graph is "built" out of the graph of x5 by selecting the Examples in the following applet:

    4. k(x) = -(1 - x)4 + 2
      Solution
      We can make the table:
      Corresponding
      Algebraic formula      		 x4 (1 + x)4 (1 - x)4   -(1 - x)4 -3(1 - x)4 + 2
      Geometric Effect Take the 4th 
      power function      		 Shift left 1 Flip horizontally Flip vertically Shift up 2
      Numerical Effect   Subtract 1 Negate Raise to the 
      4th power      		 Negate Add 2
      Numerical 
      Results      		 Inputs Outputs
      -1 -2 2 1 -1 1
      0 -1 1 0 0 2
      1 0 0 1 -1 1
      This graph is like a square graph, but steeper and flatter:

       

      You can see how this graph is "built" out of the graph of 3x by selecting the Examples in the following applet:

    Back to Exercises.

  3. Fill-in-the-blanks to create a formula for a power function, then use our transformational graphing technique to graph it.

    y = __(__x + __)__ + __

    Repeat this Exercise as often as necessary until you are confident in your ability to plot power graphs.
    Solution
    Use the Multigraph Utility in XFunctions to make a plot to check your work:

    Check that the points that you computed on your graph are truly on the graph.

    Back to Exercises.

Proportions

  1. Translate the following verbal descriptions, using the terminology of proportions, into functional equations relating the indicated variables.
    1. The surface area, A, of a cube is directly proportional to the square of the length, x, of one side.
      Solution
      A = kx2.
    2. The resistance, R, of a piece of wire is directly proportional to its length, L, and inversely proportional to the square of its diameter, d.
      Solution
      R = kL/d2

    Back to Exercises.

  2. Translate the following functional equations relating the given variables into verbal descriptions, using the terminology of proportions. 
    1. V = kr3
      Solution
      V is directly proportional to r3.
    2. y = k/x4
      Solution
      y is inversely proportional to x4.
    3. h = kV/r2
      Solution
      h is directly proportional to V and inversely proportional to r2.

    Back to Exercises.

  3. Follow the given instructions to investigate the given applied situation. 
    1. The speed, s (in mph.), you travel in your car (in fourth gear) is directly proportional to speed of your engine, r (in thousands of rpm.), indicated on your tachometer.  Give an expression for s as a function r, with a constant of proportionality, k.
      Solution
      s = kr.
    2. At one point, you notice that at 50 mph., your tachometer reads 30,000 rpm.  Use this information to solve for k.
      Solution
      Plugging in these values gives 50 = k30, so k = 5/3.
    3. If your tachometer reads 24,000 rpm., use your equation to predict your speed, s.
      Solution
      Plugging in our values for k and r gives 50 = k30, so s = (5/3)24 = 40 mph.

    Back to Exercises.

Go to Polynomials and Roots.

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