1, Unit 6 - Exponential Models
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.
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:
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:
- 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.
|Population (in millions)