Course 1, Unit 6 - Exponential Models

Summary
In the "Exponential Models" unit, students analyze situations that can be modeled well by rules of the form y = a(b)x. They construct and use data tables, graphs, and equations in the form y = a(b)x to describe and solve problems about exponential relationships such as population growth, investment of money, and decay of medicines and radioactive materials.

Key Ideas from Course 1, Unit 6

  • Exponential growth or decay relationship: In the rule y = a(b)x, b is the constant growth or decay factor. In tables where x is increasing in uniform steps, the ratios of succeeding y values will always be b. If b is greater than 1, the pattern will be exponential growth; if b is between 0 and 1, the pattern will be exponential decay. The value of a indicates the y-intercept (0, a) of the graph of the relationship.
    Example 1: y = 4(1.3)x represents an exponential growth relationship between x and y, where the initial value of y (when x = 0) is 4, and the y values increase by 30% for each increase of 1 in x values. The table would be as follows:

    x
    0
    1
    2
    3
    y
    4
    5.2
    6.76
    8.788

    Notice that each y value is 130% of the preceding y value.
    Example 2: y = 4(0.5)x represents an exponential decay relationship between x and y, where the initial value of y (when x = 0) is 4, and the y values decrease by 50% for each increase of 1 in x values. The table would begin as follows:

    x
    0
    1
    2
    3
    y
    4
    2
    1
    0.5

  • Asymptote: The graph of an exponential relationship will be asymptotic to the x-axis, getting closer and closer to the axis without ever touching or crossing it.


  • NOW-NEXT equations: Since exponential growth involves repeated multiplication by a constant factor, those patterns can be represented by equations in the general form NEXT = b * NOW, starting at a. For example, the pattern of change in a population growing at a rate of 20% per year from a base of 5 million in the year 2000 can be expressed as NEXT = 1.20NOW, starting at 5.

    Year
    2000
    2001
    2002
    2003
    Population (in millions)
    5
    6
    7.2
    8.64
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