Graph the given function by making a table of coordinates. [tex]f(x)=\left(\frac{2}{3}\right)^x[/tex] Complete the table of coordinates. | x | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | y | [tex]\frac{9}{4}[/tex] | $\square$ | 1 | [tex]\frac{2}{3}[/tex] | [tex]\frac{4}{9}[/tex] | (Type integers or fractions. Simplify your answers.) Choose the correct graph below.
Answer (2)
Calculate the missing y-value by substituting x = − 1 into the function: f ( − 1 ) = ( 3 2 ) − 1 .
Simplify the expression: f ( − 1 ) = 2 3 .
The completed table includes the point ( − 1 , 2 3 ) .
The function is an exponential decay function, and the final answer is 2 3 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( 3 2 ) x and a table with x values and corresponding y values, where y = f ( x ) . We need to find the missing y value when x = − 1 .
Substituting x = -1 To find the missing y value, we substitute x = − 1 into the function: f ( − 1 ) = ( 3 2 ) − 1
Simplifying the Expression Recall that a negative exponent means we take the reciprocal of the base: f ( − 1 ) = ( 3 2 ) − 1 = 2 3
Completing the Table So, the missing y value is 2 3 . The completed table is:
x
-2
-1
0
1
2
y
4 9
2 3
1
3 2
9 4
Identifying the Graph Since the base 3 2 is between 0 and 1, the function is an exponential decay function. This means that as x increases, y decreases. The graph should pass through the points ( − 2 , 4 9 ) , ( − 1 , 2 3 ) , ( 0 , 1 ) , ( 1 , 3 2 ) , and ( 2 , 9 4 ) .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a savings account that earns compound interest, the amount of money you have over time can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled by an exponential function.
To find the missing value for f ( − 1 ) in the function f ( x ) = ( 3 2 ) x , we calculate it to be 2 3 . The completed table of coordinates is as follows: f ( x ) values correspond to the x-values of -2, -1, 0, 1, and 2. The graph shows the decay characteristic of the function as x increases.
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