Exponential Functions

When you have finished this lesson:

• Identify the characteristics and general formula of an exponential function
• Use exponential functions to calculate values of the function for given inputs
• Given two points on an exponential function define the equation for that function (points may be given as points, a graph or a table)
• Identify whether a function is linear or exponential
• Solve exponential equations graphically

Start of Year	Co. A Salary	Co. B Salary
(Time)	Linear	Exponential
1	$40,000	$40,000
2	$42,000	$42,000
3	$44,000	44,100
4	$46,000	46,305
5	$48,000	48,620
6	$50,000	51,051

Form of an exponential function - An exponential function has the form Q(t) = ab^t, where a and b are constants, b is greater than zero and t is a variable. Exponential functions are used to model or describe processes that grow at a constant percent rate. This is in contrast to a linear process that grows by a constant amount. For example, assume that you have two job offers. Both pay $40,000 per year to start. Company A says they will give you a $2,000 raise each year for the next 5 years. Company B says they will give you a 5% raise each year for the next 5 years. The constant amount raise represents a linear process while the 5% raise represents an exponential process. The table on the right shows the difference. As you can see from the table at the end of the first year (start of year 2) your raise would be $2,000 no matter which job you choose. However, for the next year Company A would again give you a $2,000 increase while Company B would give you 5% of $42,000 or $2,100. Each year thereafter your Comapny B raise would grow because the 5% constant rate would be applied to a larger number. Thus by the start of year 6 your cumulative salary would be $1,051 dollars more if you selected Company B.

Let's now move on to the processing of determining a possible equation for a curve when you are given two points.

Exponential equations: y = ab^x

1. First case, one of the points is on the y axis with coordinates (0, y)

   • the value of a is equal to the value of the y coordinate of the point.
   • use the (x,y) pair of the other point in the formula. The only unknown is b, so solve for it.
   • rewrite the equation using the values for a and b that you just found.

   Example: put an exponential equation through the points (0, 5) and (3, 40)
   • The first point is on the y axis so the y coordinate 5 is equal to a

     5 = ab⁰

     b⁰ = 1 so 5 = a

   • use the other point: 40 = 5b³

     b³ = 40/5 = 8

   • This means that b is the cube root of 8 which is 2
   • the formula is y = 5(2)^x
   • If we want the value of the function when x = 3 it is given by y = 5(2)³ = 5(8) = 40

2. Second case, neither of the points are on the y axis.

   • take the ratio of the y values for the two points and set it equal to ab^x using the value of x for each point.
   • the a's cancel and you are left with the ratio of the y's (which are numbers) equal to the ratio of b to two different powers. Solve for b.
   • rewrite the equation using the value of b you just calculated and the x and y values of one of the points. The only unknown in the equation will be a. Solve for a.
   • write the equation of the function y = ab^x

Example: Put an exponential equation through the points (2, 45) and (4, 405)

1. take the ratio of the y values and solve for b (the a’s cancel)
2. 405/45 = b⁴/b²
3. 9 = b², solving for b; b = + or - 3, but b>0 so only +3 works
4. rewrite the equation using your value of b and one of the points, we will use the point (2, 45)

   • 45 = a(3)²
   • 45 = a (9)
   • a = 45/9 = 5

5. write the final equation using the values of a and b found above y = 5(3)^x
6. If we want the value of the function when x = 2 we calculate it as y = 5(3)² = 5 (9) = 45

Exponential Graphs and Concavity

Four graphs of exponential functions are shown on the right. The graphs depict the four possible behaviors of an exponential function.

1. A graph can be concave up and increasing such as the first figure. In this case the graph bends upwards and the rate of change of the function is increasing (it is getting steeper). Such a function is described as being concave up and increasing.
2. The second graph on the right is decreasing at a decreasing rate, meaning it is decreasing but the rate of change is slowing (slope is getting less negative)
3. The third graph is concave down and increasing. The graph is increasing but the rate of growth is slowing, the curve is getting less steep (slope is getting less positive)
4. The final graph on the right is concave down and decreasing. The graph is decreasing and the rate of decrease is growing, the curve is getting more steep (slope is getting more negative)