Consider the table below:
Which is the second coordinate of the point on the line?
Answer (1)
The problem provides a table of x and f(x) values and asks for the second term of the exponential function. We assume the function is of the form f ( x ) = a b x . We use the points (-1, 18) and (2, 2/3) to solve for a and b. We find that b = 3 1 and a = 6 . Thus, the function is f ( x ) = 6 ( 3 1 ) x . The second term, corresponding to x=1, is f ( 1 ) = 2 . Therefore, the final answer is 2 .
Explanation
Analyze the problem The table gives values of a function f(x) for certain x values. We are given the following data:
f ( − 1 ) = 18 f ( 1 ) = 6 f ( 1 ) = 2 f ( 2 ) = 3 2
The question asks for the second term of the exponential function that fits the numerical data.
Assume exponential form Let's assume the exponential function is of the form f ( x ) = a b x . We will use the given data points to create a system of equations.
Solve for b Using f ( − 1 ) = 18 and f ( 2 ) = 3 2 , we have the equations:
a b − 1 = 18 (1) a b 2 = 3 2 (2)
Dividing equation (2) by equation (1), we get:
a b − 1 a b 2 = 18 3 2 b 3 = 3 2 ⋅ 18 1 = 27 1 b = 3 27 1 = 3 1
Solve for a Substitute b = 3 1 into equation (1):
a ( 3 1 ) − 1 = 18 3 a = 18 a = 6
Find f(1) So the exponential function is f ( x ) = 6 ( 3 1 ) x .
Now, let's find the value of the function at x = 1 :
f ( 1 ) = 6 ( 3 1 ) 1 = 6 ⋅ 3 1 = 2
Determine the second term Since the question asks for the 'second term', it is likely referring to the value of the function at x = 1 . Based on our calculation and the given data, f ( 1 ) = 2 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a bacterial culture starts with 100 cells and doubles every hour, the population can be modeled by the exponential function P ( t ) = 100 ⋅ 2 t , where t is the time in hours. Understanding exponential functions allows us to predict the population size at any given time.
