Option D, g of x equals open parenthesis 3 close parenthesis open parenthesis 2 raised to the power of x close parenthesis, \( g(x)=(3)(2^x), \) should be selected.
Sketch the Points
You can make a rough sketch of g of x \( g(x) \) by plotting the four points on a coordinate grid:
From the sketch, you can eliminate options A and B because those are linear functions. A straight line connecting the points at coordinates 0, 3 \( (0,3) \) and coordinates 1, 6 \( (1,6) \) does not go through the point at coordinates 2, 12. \( (2, 12). \) Therefore, g of x \( g(x) \) is not a linear function.
Is it a linear function?
You know the function isn't linear both from the graph and from the fact that there is not a constant rate of change. The rate of change in g of x \( g(x) \) is not constant, because the change in g of x \( g(x) \) from x = 1 to x = 2 is 6, while the change from x = 2 to x = 3 is 12. You can therefore eliminate options A and B, because those are both linear functions.
Is it a quadratic function?
You can eliminate the quadratic function, option C (g of x equals 3 x squared), \( (g(x)=3x^2), \) by inspecting the points that are on the function. If this were truly the function for g of x, \( g(x), \) the point at x = 1 would be at 3, not 6.3 x squared equals open parenthesis 3 close parenthesis open parenthesis 1 close parenthesis squared = open parentheses 3 close parenthesis open parenthesis 1 close parenthesis equals 3, which is not equal to 6
$$ 3 x^2 = (3)(1)^2 = (3)(1) = 3 \ne 6 $$
Is it an exponential function?
The exponential function presented in option D is the only one left after eliminating the linear functions in options A and B and the quadratic function in option C. But you can determine if it's the correct answer by substituting the values for x from the table to ensure you get the given values for g of x \( g(x). \)g of x equals open parenthesis 3 close parenthesis open parenthesis 2 to the power of x close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 2 to the zeroth power close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 1 close parenthesis equals 3
$$ g(x) = (3)(2^x) = (3)(2^0) = (3)(1) = 3 $$
g of x equals open parenthesis 3 close parenthesis open parenthesis 2 to the power of x close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 2 to the first power close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 2 close parenthesis equals 6
$$ g(x) = (3)(2^x) = (3)(2^1) = (3)(2) = 6 $$
g of x equals open parenthesis 3 close parenthesis open parenthesis 2 to the power of x close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 2 squared close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 4 close parenthesis equals 12
$$ g(x) = (3)(2^x) = (3)(2^2) = (3)(4) = 12 $$
g of x equals open parenthesis 3 close parenthesis open parenthesis 2 to the power of x close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 2 cubed close parenthesis equals open parenthesis 3 close parenthesis open parenthesis 8 close parenthesis equals 24
$$ g(x) = (3)(2^x) = (3)(2^3) = (3)(8) = 24 $$
At least for the four points given, option D (the exponential function) is consistent and is the only function shown in the options that works for all four points.
Correct Answer
Correct Answer
Option C (n equals one thousand open parenthesis 8 close parenthesis raised to the power of t) \( (n = 1{,}000(8)^t) \) should be selected.
Explanation
To represent the number of bacteria in the culture at hour t, use an exponential growth model. In this case, the growth factor is provided in the problem: "Each hour, the number of bacteria in the culture was eight times the number that was present at the start of the preceding hour." The growth factor is 8. Call that g.
At t equals 0 \( t=0 \) hours, the number of bacteria present in the culture was 1,000. Call that s for "starting population."
The general form of an exponential growth model is the expressed by the formulaP of t equals s times g raised to the power of t,
$$ p(t) = s \cdot g^t, $$
where p(t) is the population of bacteria after t hours and t is the number of hours since t equals 0. \( t=0. \)
Therefore, substituting in the values from the problem:N equals s times g raised to the power of t
$$ n = s \cdot g^t = 1{,}000 \cdot 8^t $$
This page shows questions in the Algebra I Functions - Linear, Quadratic, and Exponential Models public release module at MSDE. Algebra 1 "Algebra I Functions - Linear, Quadratic, and Exponential Models"
Select from the list to explore. Read any associated passages and then interact with the questions here.
Copyright 2023, New Meridian Corporation. Reprinted with permission. All rights reserved. The printing, copying, redistribution, or retransmission of this content without express written permission is prohibited.
This is a multiple choice question that allows you to select only one option.
The number of bacteria initially present in a culture was 1,000. Each hour, the number of bacteria in the culture was eight times the number that was present at the start of the preceding hour, as shown in the table.
Time (hours)
Number of Bacteria in the Culture
0
1,000
1
8,000
2
64,000
3
512,000
Which equation could be used to determine n, the number of bacteria in the culture at hour t?
Copyright 2023, New Meridian Corporation. Reprinted with permission. All rights reserved. The printing, copying, redistribution, or retransmission of this content without express written permission is prohibited.
