8.6 Exponential and Logarithmic Functions, Graphs, and Models
Consider the exponential function f(x)=3(2)^x.
Fill in the table to compute f(x) for various values of x. Then graph the function.
x f(x)
0
1
2
3
-1
-2
x f(x)
0 3(2)^0 = 3(1) = 3
1 3(2)^1 = 3(2) = 6
2 3(2)^2 = 3(4) = 12
3 3(2)^3 = 3(8) = 24
-1 3(2)^(-1) = 3(1/2) = 1.5
-2 3(2)^(-2) = 3(1/4) = 0.75
To graph the function, plot the points (0, 3), (1, 6), (2, 12), (3, 24), (-1, 1.5), and (-2, 0.75) on a graph and connect them with a smooth curve.
To compute the values of f(x) for various values of x, we can substitute the values of x into the given exponential function f(x) = 3(2)^x.
x f(x)
0 3(2)^0 = 3(1) = 3
1 3(2)^1 = 3(2) = 6
2 3(2)^2 = 3(4) = 12
3 3(2)^3 = 3(8) = 24
-1 3(2)^(-1) = 3(1/2) = 3/2 = 1.5
-2 3(2)^(-2) = 3(1/4) = 3/4 = 0.75
Now let's graph the function f(x) = 3(2)^x with the computed values.
The table can be represented on a graph as follows:
(x, f(x)):
(0, 3),
(1, 6),
(2, 12),
(3, 24),
(-1, 1.5),
(-2, 0.75),
Now plot these points on a graph and connect them with a smooth curve.
The graph should show an exponential growth curve.
To compute the values of f(x) for various values of x for the exponential function f(x) = 3(2)^x, we need to substitute each value of x into the function and calculate the corresponding f(x).
Let's start by filling in the table:
x f(x)
0 3(2)^0
1 3(2)^1
2 3(2)^2
3 3(2)^3
-1 3(2)^(-1)
-2 3(2)^(-2)
To evaluate the function, we need to know that any number raised to the power of 0 is equal to 1. Therefore, we can simplify the calculations as follows:
x f(x)
0 3(1) = 3
1 3(2) = 6
2 3(4) = 12
3 3(8) = 24
-1 3(1/2) = 3/2 = 1.5
-2 3(1/4) = 3/4 = 0.75
Now that we have computed the values of f(x), we can proceed to graph the function.
To graph the function f(x) = 3(2)^x, we need to plot the values of (x, f(x)) on a coordinate plane.
The x-coordinates will be the values from the x-column in the table, and the y-coordinates will be the values from the f(x)-column.
Plotting the points on the coordinate plane, we get:
(0, 3)
(1, 6)
(2, 12)
(3, 24)
(-1, 1.5)
(-2, 0.75)
Now, connect the points with a smooth curve to represent the graph of the exponential function f(x) = 3(2)^x.
It is important to note that the graph will not pass through the points with negative x-values. As x approaches negative infinity, the function approaches 0. However, we can still plot the points and observe the overall shape and behavior of the function.