Question
An expression is shown below:
the square root of 32 plus the square root of 2
Which statement is true about the expression?
a It is rational and equal to 4 times the square root of 2.
b It is irrational and equal to 4 times the square root of 2.
c It is rational and equal to 5 times the square root of 2.
d It is irrational and equal to 5 times the square root of 2.
Question 2(Multiple Choice Worth 1 points)
(03.03 MC)
Use the graph representing bacteria decay to estimate the domain of the function and solve for the average rate of change across the domain.
An exponential function titled Bacteria Decay with x axis labeled Time, in Minutes, and y axis labeled Amount of Bacteria, in Thousands, decreasing to the right with a y intercept of 0 comma 60 and an x intercept of 18 comma 0.
a 0 ≤ x ≤ 18, −3.33
b 0 ≤ x ≤ 18, −0.3
c 0 ≤ y ≤ 60, −3.33
d 0 ≤ y ≤ 60, −0.3
Question 3(Multiple Choice Worth 1 points)
(03.04 MC)
The population f(x), in millions, of State A of a country after x years is represented by the function shown below:
f(x) = (1.08)x
The graph shows the population g(x), in millions, of State B of the country after x years:
graph of exponential function g of x that curves up from left to right and goes through points 0 comma 2 and 9 comma 4
Which conclusion is correct about the populations of State A and State B?
a The original population of State B was half the original population of State A.
b The original population of State A was half of the original population of State B.
c The original population of State B was one-fourth the original population of State A.
d The original population of State A was one-fourth the original population of State B.
Question 4(Multiple Choice Worth 1 points)
(03.02 MC)
The price of gold has increased by 35% per year from 2000. In the year 2000, Harry bought a gold ring for $590. Which of the following equations can be used to represent the price of the ring x years after 2000?
a y = 590(1.35)x
b y = 590(0.65)x
c y = 35(0.41)x
d y = 35(1.59)x
Question 5(Multiple Choice Worth 1 points)
(03.03 LC)
Given the exponential function f(x) = 54(0.45)x, classify the function as exponential growth or decay and determine the percent rate of growth or decay.
a Exponential decay, 55% decrease
b Exponential growth, 45% increase
c Exponential decay, 45% decrease
d Exponential growth, 55% increase
Question 6(Multiple Choice Worth 1 points)
(03.04 LC)
Graph the function f(x) = 5(0.21)x.
a an exponential graph decreasing from the left and crossing the y axis at 0 comma 21
b an exponential graph decreasing from the left and crossing the y axis at 0 comma 5
c an exponential graph decreasing from the left and crossing the y axis at 0 comma 18
d an exponential graph decreasing from the left and crossing the y axis at 0 comma 8
Question 7(Multiple Choice Worth 1 points)
(03.02 LC)
The function f(x) = 3,267(1 + 0.02)x represents the amount of money in a savings account where x represents time in years. What does the 3,267 represent?
a The amount of money in the savings account after one year
b The amount of money added to the savings account each year
c The initial amount of money placed in the savings account
d The growth rate
Question 8(Multiple Choice Worth 1 points)
(03.02 MC)
Initially, there were only 86 weeds in the garden. The weeds grew at a rate of 8% each week. The following function represents the weekly weed growth: f(x) = 86(1.08)x. Rewrite the function to show how quickly the weeds grow each day.
a f(x) = 86(1.08)7x; grows approximately at a rate of 5.6% daily
b f(x) = 86(1.087)x; grows approximately at a rate of 0.56% daily
c f(x) = 86(1.01)x; grows approximately at a rate of 0.1% daily
d f(x) = 86(1.01)7x; grows approximately at a rate of 1% daily
Question 9(Multiple Choice Worth 1 points)
(03.05 LC)
A ball is dropped from a certain height. The function below represents the height f(n), in feet, to which the ball bounces at the nth bounce:
f(n) = 9(0.7)n
What does the number 9 in the function represent?
a Number of times the ball was dropped
b Height from which the ball was dropped
c Height to which the ball will bounce at the nth bounce
d Number of times the ball bounces at the nth bounce
Question 10(Multiple Choice Worth 1 points)
(03.01 LC)
Simplify the square root of 5 times the cube root of 5.
a five to the five sixths power
b five to the one sixth power
c five to the two thirds power
d five to the seven sixths power
Question 11(Multiple Choice Worth 1 points)
(03.03 MC)
In the month of October, the population of a species of fish in a river increased by a factor of 1.05 every day. The function below shows the number of fish in the river f(x) after x days:
f(x) = 100(1.05)x
Which of the following is a reasonable domain for the function?
a 0 ≤ x ≤ 100
b 0 ≤ x ≤ 31
c All positive integers greater than 100
d All positive integers less than 100
Question 12(Multiple Choice Worth 1 points)
(03.05 MC)
Graph f(x) = 28(1.16)x. What is the constant percent rate of change of f(x) with respect to x? Does the graph represent growth or decay?
a 16% decay
b 16% growth
c 84% growth
d 84% decay
Question 13(Multiple Choice Worth 1 points)
(03.04 MC)
Graph the function y = 12 • 2x. Does the function represent growth or decay? What is the equation of the asymptote?
a Growth; x = 2
b Growth; y = 0
c Decay; x = 2
d Decay; y = 0
Question 14(Multiple Choice Worth 1 points)
(03.05 MC)
Albert invested money into the stock market, and the table represents his earnings. What type of function could be used to model his bank account as a function of time? Justify your answer.
Week Balance ($)
1 1,426
2 1,528
3 1,630
4 1,732
a This is a linear function because there is a common difference in the balance between the weeks.
b This is a linear function because there is a common ratio in the balance between the weeks.
c This is an exponential function because there is a common difference in the balance between the weeks.
d This is an exponential function because there is a common ratio in the balance between the weeks.
Question 15(Multiple Choice Worth 1 points)
(03.05 MC)
Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop, and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $69,682.50 $72,429.27
2000 $78,943.50 $79,967.77
2005 $88,204.50 $88,290.88
2006 $90,056.70 $90,056.70
2007 $91,908.90 $91,857.83
2010 $97,465.50 $97,480.27
a f(x) is exponential; an exponential function increases more slowly than a linear function.
b f(x) is exponential; f(x) increased more overall than g(x).
c g(x) is exponential; g(x) has a higher starting value and higher ending value.
d g(x) is exponential; an exponential function increases faster than a linear function
the square root of 32 plus the square root of 2
Which statement is true about the expression?
a It is rational and equal to 4 times the square root of 2.
b It is irrational and equal to 4 times the square root of 2.
c It is rational and equal to 5 times the square root of 2.
d It is irrational and equal to 5 times the square root of 2.
Question 2(Multiple Choice Worth 1 points)
(03.03 MC)
Use the graph representing bacteria decay to estimate the domain of the function and solve for the average rate of change across the domain.
An exponential function titled Bacteria Decay with x axis labeled Time, in Minutes, and y axis labeled Amount of Bacteria, in Thousands, decreasing to the right with a y intercept of 0 comma 60 and an x intercept of 18 comma 0.
a 0 ≤ x ≤ 18, −3.33
b 0 ≤ x ≤ 18, −0.3
c 0 ≤ y ≤ 60, −3.33
d 0 ≤ y ≤ 60, −0.3
Question 3(Multiple Choice Worth 1 points)
(03.04 MC)
The population f(x), in millions, of State A of a country after x years is represented by the function shown below:
f(x) = (1.08)x
The graph shows the population g(x), in millions, of State B of the country after x years:
graph of exponential function g of x that curves up from left to right and goes through points 0 comma 2 and 9 comma 4
Which conclusion is correct about the populations of State A and State B?
a The original population of State B was half the original population of State A.
b The original population of State A was half of the original population of State B.
c The original population of State B was one-fourth the original population of State A.
d The original population of State A was one-fourth the original population of State B.
Question 4(Multiple Choice Worth 1 points)
(03.02 MC)
The price of gold has increased by 35% per year from 2000. In the year 2000, Harry bought a gold ring for $590. Which of the following equations can be used to represent the price of the ring x years after 2000?
a y = 590(1.35)x
b y = 590(0.65)x
c y = 35(0.41)x
d y = 35(1.59)x
Question 5(Multiple Choice Worth 1 points)
(03.03 LC)
Given the exponential function f(x) = 54(0.45)x, classify the function as exponential growth or decay and determine the percent rate of growth or decay.
a Exponential decay, 55% decrease
b Exponential growth, 45% increase
c Exponential decay, 45% decrease
d Exponential growth, 55% increase
Question 6(Multiple Choice Worth 1 points)
(03.04 LC)
Graph the function f(x) = 5(0.21)x.
a an exponential graph decreasing from the left and crossing the y axis at 0 comma 21
b an exponential graph decreasing from the left and crossing the y axis at 0 comma 5
c an exponential graph decreasing from the left and crossing the y axis at 0 comma 18
d an exponential graph decreasing from the left and crossing the y axis at 0 comma 8
Question 7(Multiple Choice Worth 1 points)
(03.02 LC)
The function f(x) = 3,267(1 + 0.02)x represents the amount of money in a savings account where x represents time in years. What does the 3,267 represent?
a The amount of money in the savings account after one year
b The amount of money added to the savings account each year
c The initial amount of money placed in the savings account
d The growth rate
Question 8(Multiple Choice Worth 1 points)
(03.02 MC)
Initially, there were only 86 weeds in the garden. The weeds grew at a rate of 8% each week. The following function represents the weekly weed growth: f(x) = 86(1.08)x. Rewrite the function to show how quickly the weeds grow each day.
a f(x) = 86(1.08)7x; grows approximately at a rate of 5.6% daily
b f(x) = 86(1.087)x; grows approximately at a rate of 0.56% daily
c f(x) = 86(1.01)x; grows approximately at a rate of 0.1% daily
d f(x) = 86(1.01)7x; grows approximately at a rate of 1% daily
Question 9(Multiple Choice Worth 1 points)
(03.05 LC)
A ball is dropped from a certain height. The function below represents the height f(n), in feet, to which the ball bounces at the nth bounce:
f(n) = 9(0.7)n
What does the number 9 in the function represent?
a Number of times the ball was dropped
b Height from which the ball was dropped
c Height to which the ball will bounce at the nth bounce
d Number of times the ball bounces at the nth bounce
Question 10(Multiple Choice Worth 1 points)
(03.01 LC)
Simplify the square root of 5 times the cube root of 5.
a five to the five sixths power
b five to the one sixth power
c five to the two thirds power
d five to the seven sixths power
Question 11(Multiple Choice Worth 1 points)
(03.03 MC)
In the month of October, the population of a species of fish in a river increased by a factor of 1.05 every day. The function below shows the number of fish in the river f(x) after x days:
f(x) = 100(1.05)x
Which of the following is a reasonable domain for the function?
a 0 ≤ x ≤ 100
b 0 ≤ x ≤ 31
c All positive integers greater than 100
d All positive integers less than 100
Question 12(Multiple Choice Worth 1 points)
(03.05 MC)
Graph f(x) = 28(1.16)x. What is the constant percent rate of change of f(x) with respect to x? Does the graph represent growth or decay?
a 16% decay
b 16% growth
c 84% growth
d 84% decay
Question 13(Multiple Choice Worth 1 points)
(03.04 MC)
Graph the function y = 12 • 2x. Does the function represent growth or decay? What is the equation of the asymptote?
a Growth; x = 2
b Growth; y = 0
c Decay; x = 2
d Decay; y = 0
Question 14(Multiple Choice Worth 1 points)
(03.05 MC)
Albert invested money into the stock market, and the table represents his earnings. What type of function could be used to model his bank account as a function of time? Justify your answer.
Week Balance ($)
1 1,426
2 1,528
3 1,630
4 1,732
a This is a linear function because there is a common difference in the balance between the weeks.
b This is a linear function because there is a common ratio in the balance between the weeks.
c This is an exponential function because there is a common difference in the balance between the weeks.
d This is an exponential function because there is a common ratio in the balance between the weeks.
Question 15(Multiple Choice Worth 1 points)
(03.05 MC)
Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop, and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $69,682.50 $72,429.27
2000 $78,943.50 $79,967.77
2005 $88,204.50 $88,290.88
2006 $90,056.70 $90,056.70
2007 $91,908.90 $91,857.83
2010 $97,465.50 $97,480.27
a f(x) is exponential; an exponential function increases more slowly than a linear function.
b f(x) is exponential; f(x) increased more overall than g(x).
c g(x) is exponential; g(x) has a higher starting value and higher ending value.
d g(x) is exponential; an exponential function increases faster than a linear function
Answers
GPT-4o mini
Here's a breakdown of the answers to your questions:
### Question 1
The expression is: \(\sqrt{32} + \sqrt{2}\).
\[
\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}
\]
Thus, the expression becomes:
\[
4\sqrt{2} + \sqrt{2} = 5\sqrt{2}
\]
**Answer:** d It is irrational and equal to 5 times the square root of 2.
### Question 2
The average rate of change from 0 to 18 (where the graph starts at 60 and ends at 0):
\[
\text{Average Rate of Change} = \frac{0 - 60}{18 - 0} = -\frac{60}{18} = -3.33
\]
**Answer:** a 0 ≤ x ≤ 18, −3.33
### Question 3
The original population of State A at \(x = 0\) (when \(x\) is in years):
\[
f(0) = (1.08)^0 = 1 \text{ million}
\]
For State B, it’s not provided, but given points through which it passes and considering it doubles:
At \(x = 0\), \(g(0) = 2\) million. Hence, the original population of State B was 2 million, and the population of State A was 1 million.
**Answer:** a The original population of State B was half the original population of State A.
### Question 4
Since the price increases by 35%, the equation is:
\[
y = 590(1.35)^x
\]
**Answer:** a y = 590(1.35)x
### Question 5
The function is \(f(x) = 54(0.45)^x\). The base \(0.45\) indicates a decay; hence the percent is:
\[
100\% - 45\% = 55\%
\]
**Answer:** a Exponential decay, 55% decrease
### Question 6
The function \(f(x) = 5(0.21)^x\) crosses the y-axis at \(x = 0\):
\[
f(0) = 5(0.21)^0 = 5
\]
**Answer:** b an exponential graph decreasing from the left and crossing the y axis at 0 comma 5
### Question 7
The number 3,267 represents the initial amount of money placed in the savings account at \(x=0\).
**Answer:** c The initial amount of money placed in the savings account
### Question 8
To find the daily growth rate from 8% weekly:
The daily rate \(r\) is given by:
\[
1.08 = (1 + r)^7 \quad \Rightarrow \quad r \approx 0.0106 \Rightarrow 1.06\%
\]
**Answer:** a f(x) = 86(1.08)7x; grows approximately at a rate of 5.6% daily
### Question 9
In the function \(f(n) = 9(0.7)^n\), the number 9 represents the height from which the ball is dropped.
**Answer:** b Height from which the ball was dropped
### Question 10
To simplify \( \sqrt{5} \times \sqrt[3]{5} \):
\[
= 5^{1/2} \cdot 5^{1/3} = 5^{(3/6 + 2/6)} = 5^{5/6}
\]
**Answer:** a five to the five sixths power
### Question 11
A reasonable domain for the function would be:
\( \text{Days in October} = 31 \)
**Answer:** b 0 ≤ x ≤ 31
### Question 12
The function \(f(x) = 28(1.16)^x\) represents a constant growth of \(16\%\).
**Answer:** b 16% growth
### Question 13
The function \(y = 12 \cdot 2^x \) shows growth since the base is greater than 1 and the horizontal asymptote is \(y = 0\).
**Answer:** b Growth; y = 0
### Question 14
The balances indicate a sequence where there is a common difference indicating it is a linear function.
**Answer:** a This is a linear function because there is a common difference in the balance between the weeks.
### Question 15
Looking at the balance data, if the increases aren't constant and seem to accelerate, the garden shop profits \(f(x)\) would represent exponential growth.
**Answer:** a f(x) is exponential; an exponential function increases more slowly than a linear function.
I hope this helps you with your questions! Let me know if you need any more assistance.
### Question 1
The expression is: \(\sqrt{32} + \sqrt{2}\).
\[
\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}
\]
Thus, the expression becomes:
\[
4\sqrt{2} + \sqrt{2} = 5\sqrt{2}
\]
**Answer:** d It is irrational and equal to 5 times the square root of 2.
### Question 2
The average rate of change from 0 to 18 (where the graph starts at 60 and ends at 0):
\[
\text{Average Rate of Change} = \frac{0 - 60}{18 - 0} = -\frac{60}{18} = -3.33
\]
**Answer:** a 0 ≤ x ≤ 18, −3.33
### Question 3
The original population of State A at \(x = 0\) (when \(x\) is in years):
\[
f(0) = (1.08)^0 = 1 \text{ million}
\]
For State B, it’s not provided, but given points through which it passes and considering it doubles:
At \(x = 0\), \(g(0) = 2\) million. Hence, the original population of State B was 2 million, and the population of State A was 1 million.
**Answer:** a The original population of State B was half the original population of State A.
### Question 4
Since the price increases by 35%, the equation is:
\[
y = 590(1.35)^x
\]
**Answer:** a y = 590(1.35)x
### Question 5
The function is \(f(x) = 54(0.45)^x\). The base \(0.45\) indicates a decay; hence the percent is:
\[
100\% - 45\% = 55\%
\]
**Answer:** a Exponential decay, 55% decrease
### Question 6
The function \(f(x) = 5(0.21)^x\) crosses the y-axis at \(x = 0\):
\[
f(0) = 5(0.21)^0 = 5
\]
**Answer:** b an exponential graph decreasing from the left and crossing the y axis at 0 comma 5
### Question 7
The number 3,267 represents the initial amount of money placed in the savings account at \(x=0\).
**Answer:** c The initial amount of money placed in the savings account
### Question 8
To find the daily growth rate from 8% weekly:
The daily rate \(r\) is given by:
\[
1.08 = (1 + r)^7 \quad \Rightarrow \quad r \approx 0.0106 \Rightarrow 1.06\%
\]
**Answer:** a f(x) = 86(1.08)7x; grows approximately at a rate of 5.6% daily
### Question 9
In the function \(f(n) = 9(0.7)^n\), the number 9 represents the height from which the ball is dropped.
**Answer:** b Height from which the ball was dropped
### Question 10
To simplify \( \sqrt{5} \times \sqrt[3]{5} \):
\[
= 5^{1/2} \cdot 5^{1/3} = 5^{(3/6 + 2/6)} = 5^{5/6}
\]
**Answer:** a five to the five sixths power
### Question 11
A reasonable domain for the function would be:
\( \text{Days in October} = 31 \)
**Answer:** b 0 ≤ x ≤ 31
### Question 12
The function \(f(x) = 28(1.16)^x\) represents a constant growth of \(16\%\).
**Answer:** b 16% growth
### Question 13
The function \(y = 12 \cdot 2^x \) shows growth since the base is greater than 1 and the horizontal asymptote is \(y = 0\).
**Answer:** b Growth; y = 0
### Question 14
The balances indicate a sequence where there is a common difference indicating it is a linear function.
**Answer:** a This is a linear function because there is a common difference in the balance between the weeks.
### Question 15
Looking at the balance data, if the increases aren't constant and seem to accelerate, the garden shop profits \(f(x)\) would represent exponential growth.
**Answer:** a f(x) is exponential; an exponential function increases more slowly than a linear function.
I hope this helps you with your questions! Let me know if you need any more assistance.