Two scientists are running experiments on the same type of friction in different control groups. The results are shown in the graph using the functions f(x) and g(x):

Question

graph of two functions in quadrant 1, f of x equals 2 times x plus 3 and g of x equals one tenth times 3 to the x power

Which statement best describes the graph of f(x) and g(x)?

The graphs will both have their y-intercept equal to 3.
 The graphs will both have their y-intercept equal to 4.
 The graph of f(x) will eventually exceed the graph of g(x).
 The graph of g(x) will eventually exceed the graph of f(x).
Question 2(Multiple Choice Worth 1 points)
(03.05 MC)

Rosana is tracking how many customers she can serve in a morning. She listed the number of customers served per hour in the following table:

Hour (x)	Number of Customers
f(x)
1	3
2	5
3	7

Determine if these data represent a linear function or an exponential function, and give the common difference or ratio.
 This is a linear function because there is a common difference of 2.
 This is an exponential function because there is a common ratio of 2.
 This is a linear function because there is a common difference of 3.
 This is an exponential function because there is a common ratio of 3.
Question 3(Multiple Choice Worth 1 points)
(03.05 MC)

Claire keeps track of the miles per gallon her car gets per week. She has accumulated the following data:

(1, 22.42), (2, 22.84), (3, 23.26), (4, 23.68)

What is the common difference or ratio?

The common ratio is 0.99.
 The common ratio is 1.05.
 The common difference is 0.42.
 The common difference is 0.84.
Question 4(Multiple Choice Worth 1 points)
(03.05 MC)

The amount of a radioactive material changes with time. The table below shows the amount of radioactive material f(t) left after time t:

t(hours)	0	1	2
f(t)	100	50	25

Which exponential function best represents the relationship between f(t) and t?
 f(t) = 0.5(100)t
 f(t) = 100(0.25)t
 f(t) = 0.25(50)t
 f(t) = 100(0.5)t
Question 5(Multiple Choice Worth 1 points)
(03.05 MC)

Find the percent rate of growth for an exponential function that contains the ordered pairs (0, 30) and (1, 31.8).

0.6%
 6%
 0.06%
 1.06%
Question 6(Multiple Choice Worth 1 points)
(03.05 LC)

Maggie and her friends are each able to recruit a certain number of people each year to sample a product. The number of people per year is represented by the function f(t) = 3(2)t. What does the t represent? How many people were recruited in year 6?

t represents the year number; 36 people were recruited in year 6.
 t represents the year number; 192 people were recruited in year 6.
 t represents the number of people; 192 people were recruited in year 6.
 t represents the number of people; 36 people were recruited in year 6.
Question 7(Multiple Choice Worth 1 points)
(03.05 LC)

Antonia is keeping track of the number of pages she reads. She started the school year having already read 95 pages of her book, and she then reads 20 pages each day. The function for her total pages read is f(x) = 20x + 95. What do f(x) and x represent in Antonia's situation?

f(x) represents the number of days; x represents the total number of pages Antonia has read.
 f(x) represents the total number of pages Antonia has read; x represents the number of days.
 f(x) represents the number of pages Antonia reads per day; x represents the number of days.
 f(x) represents the total number of pages Antonia has read; x represents the number of pages she reads per day.
Question 8(Multiple Choice Worth 1 points)
(03.05 MC)

Cecelia studied for the SAT for 20 minutes every day in the first week, 30 minutes in the second week, 40 minutes in the third week, and 50 minutes in the fourth week.

Christopher studied for the SAT for 5 minutes every day in the first week, 10 minutes in the second week, 20 minutes in the third week, and 40 minutes in the fourth week.

Which statement best describes the methods used by Cecelia and Christopher to increase the time they spent studying?

Cecelia's method is exponential because the number of minutes increased by an equal number every week.
 Christopher's method is exponential because the number of minutes increased by an equal factor every week.
 Both Christopher's and Cecelia's methods are linear because the number of minutes increased by an equal factor every week.
 Both Christopher's and Cecelia's methods are linear because the number of minutes increased by an equal number every week.
Question 9(Multiple Choice Worth 1 points)
(03.05 MC)

Julia opened a new flower shop, and her daily sales are modeled by f(x) = 2(1.25)x. Determine the rate of growth.

2%
 125%
 75%
 25%
Question 10(Multiple Choice Worth 1 points)
(03.05 MC)

The balances in two separate bank accounts that grow each month at different rates are represented by the functions f(x) and g(x). In what month do the funds in the f(x) bank account exceed those in the g(x) bank account?

Month (x)	f(x) = 2x	g(x) = 4x + 12
1	2	16
2	4	20
 Month 3
 Month 4
 Month 5
 Month 6

GPT-4o mini · Answer

Let's answer each of your questions one by one.

Question 1: The functions provided are \(f(x) = 2x + 3\) and \(g(x) = 0.1 \cdot 3^x\).

The y-intercept of \(f(x)\) is \(3\) (when \(x=0\), \(f(0)=3\)).
The y-intercept of \(g(x)\) is \(0.1 \cdot 3^0 = 0.1\) (when \(x=0\)).
As \(x\) increases, \(f(x)\) will grow linearly while \(g(x)\) grows exponentially.
Therefore, the graph of g(x) will eventually exceed the graph of f(x).

Answer: The graph of g(x) will eventually exceed the graph of f(x).

Question 2: The data shows:

\(f(1) = 3\)
\(f(2) = 5\) (difference of \(2\))
\(f(3) = 7\) (difference of \(2\))

Since there is a common difference of \(2\), this represents a linear function.

Answer: This is a linear function because there is a common difference of 2.

Question 3: The data shows:

\(f(1) = 22.42\)
\(f(2) = 22.84\) (difference of \(0.42\))
\(f(3) = 23.26\) (difference of \(0.42\))
\(f(4) = 23.68\) (difference of \(0.42\))

The common difference here is \(0.42\).

Answer: The common difference is 0.42.

Question 4: The table shows a clear exponential decay as follows:

\(f(0) = 100\)
\(f(1) = 50\) (half of \(100\))
\(f(2) = 25\) (half of \(50\))

This implies the relationship can be represented with the function \(f(t) = 100(0.5)^t\).

Answer: f(t) = 100(0.5)^t.

Question 5: To find the percent rate of growth:

The increase from \(30\) to \(31.8\) is \(1.8\).
The percent increase = \(\frac{1.8}{30} \times 100% = 6%\).

Answer: 6%.

Question 6: In the function \(f(t) = 3(2)^t\):

\(t\) represents the year number.
In year \(6\), \(f(6) = 3(2)^6 = 3 \times 64 = 192\).

Answer: t represents the year number; 192 people were recruited in year 6.

Question 7: In the function \(f(x) = 20x + 95\):

\(f(x)\) represents the total number of pages Antonia has read.
\(x\) represents the number of days.

Answer: f(x) represents the total number of pages Antonia has read; x represents the number of days.

Question 8: Cecelia increases her study time in equal increments (linear) while Christopher doubles his study time each week (exponential).

Answer: Christopher's method is exponential because the number of minutes increased by an equal factor every week.

Question 9: In the function \(f(x) = 2(1.25)^x\), the growth rate is the coefficient of the exponential factor minus 1:

\(1.25 - 1 = 0.25\) or 25% growth rate.

Answer: 25%.

Question 10: To determine when \(f(x) = 2^x\) exceeds \(g(x) = 4x + 12\), evaluate both functions for increasing integer values of \(x\):

Month 1: \(f(1) = 2\), \(g(1) = 16\) (g > f)
Month 2: \(f(2) = 4\), \(g(2) = 20\) (g > f)
Month 3: \(f(3) = 8\), \(g(3) = 24\) (g > f)
Month 4: \(f(4) = 16\), \(g(4) = 28\) (g > f)
Month 5: \(f(5) = 32\), \(g(5) = 32\) (g = f)
Month 6: \(f(6) = 64\), \(g(6) = 36\) (f > g)

Thus, the funds in the \(f(x)\) bank account exceed those in the \(g(x)\) bank account in Month 6.

Answer: Month 6.