This table gives the number of AIDS cases reported in men in Los Angeles from 1983-1989:
Month and Year
of Diagnosis AIDS cases reported in period
Jan-June 83 116
July-Dec 83 154
Jan-June 84 197
July Dec 84 269
Jan-June 85 415
July-Dec 85 503
Jan-June 86 668
July-Dec 86 773
Jan-June 87 952
July-Dec 87 933
Jan-June 88 955
July-Dec 88 943
Jan-June 89 967
• Looking at the data, determine for which points an exponential function fits this data, and for which points the growth is not exponential.
• Because patterns change in real life, to create a true function to represent these data, you need to piece together two equations, called a piecewise function: you'll need to clearly state the domain for which each equation applies. Find these two equations of best fit either by hand or with a graphing calculator, making sure that you're finding two (or more) separate equations for the different sections of the graph.
To determine whether an exponential function fits the data or not, we can look at the pattern of growth. An exponential function has a constant growth rate, meaning that the data points should increase by a consistent factor each time period.
Let's first calculate the ratio between consecutive terms to see if it is approximately constant:
For the period Jan-June 83 to July-Dec 83: Ratio = 154 / 116 ≈ 1.33
For the period July-Dec 83 to Jan-June 84: Ratio = 197 / 154 ≈ 1.28
For the period Jan-June 84 to July-Dec 84: Ratio = 269 / 197 ≈ 1.36
For the period July-Dec 84 to Jan-June 85: Ratio = 415 / 269 ≈ 1.54
For the period Jan-June 85 to July-Dec 85: Ratio = 503 / 415 ≈ 1.21
For the period July-Dec 85 to Jan-June 86: Ratio = 668 / 503 ≈ 1.33
For the period Jan-June 86 to July-Dec 86: Ratio = 773 / 668 ≈ 1.16
For the period July-Dec 86 to Jan-June 87: Ratio = 952 / 773 ≈ 1.23
For the period Jan-June 87 to July-Dec 87: Ratio = 933 / 952 ≈ 0.98
For the period July-Dec 87 to Jan-June 88: Ratio = 955 / 933 ≈ 1.02
For the period Jan-June 88 to July-Dec 88: Ratio = 943 / 955 ≈ 0.99
For the period July-Dec 88 to Jan-June 89: Ratio = 967 / 943 ≈ 1.03
From these calculations, we can see that the growth rate is not consistent for the entire dataset. There is a general pattern of growth, but there are also periods where the growth rate varies.
To create a piecewise function, we need to identify the points where the pattern of growth changes. From the data, it appears that there are two distinct patterns: an initial slower growth and a later period of faster growth.
Based on this observation, we can create a piecewise function to represent the data:
For the initial period (Jan-June 83 to Jan-June 86), we can use an exponential function of the form: f(x) = ae^(bx), where a and b are constants.
For the later period (Jan-June 86 to Jan-June 89), we can also use an exponential function of the form: g(x) = cy^x, where c and y are constants.
To find the best-fit exponential equations for each period, you can use a graphing calculator or a regression analysis tool. These tools will provide the values of the constants that optimize the fit to the data.
Once you have the constants, you can write the equations of best fit for each period, specifying the domain for which each equation applies.