Question 1(Multiple Choice Worth 1 points)
(03.03 LC)
The number of members, f(x), in a baseball club increased by 45% every year over a period of x years. The function below shows the relationship between f(x) and x:
f(x) = 25(1.45)x
Which of the following graphs best represents the function?
a. graph of exponential function going up from left to right through the point approximately 0 comma 1 and 5 tenths and approximately 4 tenths comma 5
b. graph of exponential function going up from left to right in quadrant 1 through the point 0, 0 and continuing toward infinity
c. graph of exponential function going up from left to right through the point approximately 0 comma 1 and approximately 2 comma 2
d. graph of exponential function going up from left to right through the point 0 comma 25 and approximately 3 comma 75
The correct answer is d. graph of exponential function going up from left to right through the point 0 comma 25 and approximately 3 comma 75. The function f(x) = 25(1.45)^x shows exponential growth, starting at 25 (when x = 0), and increasing by 45% each year. As x increases, the graph will increase rapidly, representing the exponential growth. The points (0, 25) and approximately (3, 75) align with this behavior.
The table represents a continuous exponential function f(x).
x 2 3 4 5
f(x) 9 3 1 one-third
Graph f(x) and identify the y-intercept.
a. 9
b. 15
c. 36
d. 81
To graph the continuous exponential function f(x), we can plot the given points (2, 9), (3, 3), (4, 1), and (5, one-third).
The graph of the exponential function would start at (2, 9) and decrease rapidly as x increases. However, we are not given specific values for the y-coordinate when x is not an integer, so we cannot accurately draw the graph beyond the given points.
To find the y-intercept, we can determine the value of f(x) when x = 0.
Using the given points, we observe that f(3) = 3 and f(4) = 1.
Since the function is continuous, we can assume that the value of f(x) decreases steadily from 3 to 1 as x increases.
Therefore, we can estimate that f(2) is likely to be greater than 9, the y-coordinate at x = 3.
From the given answer choices, the only option that fits this estimation is (c) 36.
Therefore, the y-intercept is 36.
A talent competition on television had five elimination rounds. After each elimination, only one-third of the contestants were sent to the next round. The table below shows the number of contestants in each round of the competition:
x 1 2 3 4 5
f(x) 243 81 27 9 3
Compute the average rate of change of f(x) from x = 1 to x = 4, and describe what it represents.
a. −78 contestants per round, and it represents the number of contestants who will reach the final round
b. −234 contestants per round, and it represents the number of contestants who will reach the final round
c. −234 contestants per round, and it represents the average rate at which the number of contestants changed from the first round to the fourth round
d. −78 contestants per round, and it represents the average rate at which the number of contestants changed from the first round to the fourth round
To compute the average rate of change of f(x) from x = 1 to x = 4, we can use the formula:
Average Rate of Change = (f(4) - f(1))/(4 - 1)
From the given table, we can determine that f(1) = 243 and f(4) = 9.
Substituting these values into the formula, we get:
Average Rate of Change = (9 - 243)/(4 - 1)
= (-234)/3
= -78
Therefore, the average rate of change of f(x) from x = 1 to x = 4 is -78 contestants per round.
Now, let's consider the last part of the question: what does this average rate of change represent?
Since we are computing the average rate of change from the first round to the fourth round, the -78 contestants per round represents the average rate at which the number of contestants changed from the first round to the fourth round.
Therefore, the correct answer is (d) -78 contestants per round, and it represents the average rate at which the number of contestants changed from the first round to the fourth round.
The correct answer is:
a. graph of exponential function going up from left to right through the point approximately 0,1 and approximately 0.4,5
To determine which graph best represents the function f(x) = 25(1.45)^x, we can analyze the key characteristics of exponential functions.
Exponential functions have the form f(x) = a * b^x, where "a" is the initial value and "b" is the growth factor.
In this case, the initial value, or f(0), is 25, as given by f(0) = 25(1.45)^0 = 25(1) = 25.
The growth factor, or b, is 1.45, as given in the expression f(x) = 25(1.45)^x.
Now, let's analyze the answer choices:
a. The graph goes through the points (0.1, 0.5) and (0.4, 5). However, it does not start at (0, 1) and it does not have an initial value of 25. Therefore, it is not the correct graph.
b. The graph starts at (0, 0), which is not the correct initial value of 25. Also, the y-values keep increasing towards infinity. Therefore, it is not the correct graph.
c. The graph goes through the points (0.1, 1) and (2, 2). However, it does not start at (0, 25) and it does not have an exponential growth factor of 1.45. Therefore, it is not the correct graph.
d. The graph starts at (0, 25), which is the correct initial value. Also, it goes through the point (3, 75), which shows the exponential growth. Therefore, it is the correct graph.
Therefore, the correct choice is d. graph of exponential function going up from left to right through the point 0, 25 and approximately 3, 75.