## 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