a The equation of an exponential function is y=ab^x. The given points must satisfy the equation.
B
b The b-value tells us how fast the exponential function is changing. Increasing the value of x by one multiplies the value of the function by b.
C
c Think about a situation when each time a certain interval passes, a certain amount gets multiplied by a given value.
A
a y=4(1.5)^x
B
b See solution.
C
c See solution.
Practice makes perfect
a We want to write an exponential function for the graph that passes through the given points. Let's consider the general form for this type of a function.
y=ab^x
Since we want the points to lie on the graph, they must both satisfy this equation. By substituting these points into the formula we get two equations.
Point
y = ab^x
( 1, 6)
6=ab^1
( 1, 6)
6=ab^1
( 5, 30.375)
30.375=ab^5
If we combine these equations, we get a system of equations which we can solve for a and b.
Finally, we can write the full equation of the exponential function.
y=4(1.5)^x
b The b-value from Part A tells us how fast the function increases — since b = 1.5, increasing the value of x by one makes the value of the function 1.5 times greater. Let's take a look at a following example with x = 0.
As we can see, each time the value of x was increased by 1, the value of function increased 1.5 = b times.
c Let's think of a situation that when each time a certain interval passes, the amount of something gets multiplied by a given value. One such example would be the amount of debt we owe if we borrowed $4 with 0.5 annual interest.
