Problem Set 1.1 Spring 2009


Variables, Functions, and Graphs

*  Indicates a problem that is more challenging or involved.

This problem set emphasizes practice with fractions, percentages, algebraic equations, as well as variables, functions, and graphs.  If your quantitative skills are rusty, you may wish to refer to tutorial material or introductory texts. 

[PROBLEM SET 1.1 - QUESTION #1]

  • 1.The preceding bar graph represents the average number of patients seen at a clinic over a six-month period.  Use this graph to answer the following questions:

    1. Which service receives the fewest number of patients per week?
    2. On which day do the fewest number of hearing patients visit the clinic?
    3. What fraction of the speech patients visit the clinic on Monday?
    4. What percentage of the clinic's patients are language patients?
    5. Describe any trends, or patterns, in visits to the clinic over the week
    6. What fraction of the Monday patients are speech patients?
    7. What might cause the peak in the number of speech patients on Wednesday? Use your imagination and common sense.
  • 2. Solve the following equations for x.

    a.  5(x - 1) = 16 - x b.   3y = 6(x + 7) c.   1/x  =  7 d.  2x + Hx =  -4(H + 2)
    e.   x 2  =  144 f.   2 x = 16 g.*   3 x = 1/27  

  • 3. Graph the following functions. (Hand-drawn sketches are fine, but you might want to use Excel or a graphing calculator to help you better understand the graphs.)
    a. f(x) = 6x -10  b.f(x) = -2x 2 + 6x + 3 c. f(x) = log2 x d. f(x) = 2x

    When considering problems 4, recall that a regression equation is an equation representing a line or curve that, when graphed, closely matches the graph of some data set.  Among other things, regression equations are used to estimate points that are not in the original data set.

  • 4. The table below shows population growth over time.  The figure below shows a graph of population growth along with regression lines and equations that are fitted to the raw data.  Given the table and graph, answer the following questions:
    1. What was the percentage increase in population from 1960 to 1980?
    2. Use the exponential regression equation, y = 2.1596 · 10-12 · e 0.019336x, to calculate what the estimated population will be in 2030. (This may be done using Excel or a calculator.)
    3. Solve the linear regression equation, y = 941.7818x - 1776064 , for x.  Explain the difference between the two forms of these equations (one form is y =  . . .  and the other is x = . . .).  Why might each be useful?
    4. Enter the exponential and linear equations in an Excel spreadsheet (or use a graphing calculator if you have one) and find the two points where these graphs cross. To do this in Excel, you will create three columns: one for x-values ranging from 1890 to 2030 in steps of size 1 (use the AutoFill feature or the fill-series command under the "Edit" menu), then enter the linear and exponential equations in the next two columns.  Finally, graph the two functions and find the year(s) in which they cross. 
    5. Which equation (linear or exponential) do you think is the best representation of the individual data points and the county's population growth?

    Year Population   Year Population
    1890 17,673   1950 50,080
    1900 20,873   1960 59,225
    1910 23,426   1970 84,849
    1920 24,519   1980 98,785
    1930 35,974   1990 108,978
    1940 36,534   2000 ?

    [PROBLEM SET 1.1 - QUESTION #4]

  • 5. Sketch appropriate charts or graphs for the following situations.  Use plausible values--don't agonize over details; the important part is to select a useful type of chart or graph.  (Hand-drawn sketches are fine.)
    1. The height of a person over the course of her life.
    2. The distribution of bachelor's, masters, and doctoral degree students at the University of Applied Acoustics.
    3. The way you spend your time on an average school day.
    4. The hourly average temperature during a week in July.
    5. The hourly average temperature during a week in January.  Explain the similarities and differences between this graph and the one for (d).
    6. The height of a person above the ground riding on a Ferris wheel.
  • 6. Write a one-parameter function expressing the relationship between the number of people on a train and the number of passenger cars in the train.  Explain the parameter and use the function in an example (don't spend time researching trains!).
  • * 7. Write a two-parameter function for the price of a pizza in terms of its size in inches, and the number of ingredients.  Specify reasonable values for the parameters and use the function in an example.  Hint: Some people call pizzas pizza-pies.