The exponential function f(x) = 3(5)x grows by a factor of 25 between x = 1 and x = 3. What factor does it grow by between x = 5 and x = 7?
A) 5
B) 25
C) 125
D) 625
3)
If a city that currently has a population of 1000 triples in size every 8 years, what will the population be in 24 years? Is the population growth modeled by a linear function or an exponential function?
A) 12,000; linear
B) 18,000; exponential
C) 20,000; linear
D) 27,000; exponential
4)
Consider this function for cell duplication where the cells duplicate every minute.
f(x) = 75(2)x
Determine what each parameter in the function represents.
A) The 75 is the initial number of cells, and the 2 indicates that the number of cells doubles every minute.
B) The 75 is the initial number of cells, and the 2 indicates that the number of cells increases by 2 every minute.
C) The 75 is the number of cells at 1 minute, and the 2 indicates that the number of cells doubles every minute.
D) The 75 is the number of cells at 1 minute, and the 2 indicates that the number of cells increases by 2 every minute.
5)
A certain population of bacteria has an average growth rate of 2% every five hours. The formula for the growth of the bacteria's population is A = P0 · 1.02t.
If you begin with 200 bacteria, about how many bacteria will there be after 100 hours?
A) 220
B) 297
C) 1,449
D) 5,248
6)
What is the effect on the graph of the function f(x) = 2x when f(x) is replaced with f(x −
3/5)?
A) translate vertically 3/5 units up
B) translate vertically 3/5 units down
C) translate horizontally 3\5 units left
D) translate horizontally 3/5 units right
7)
This exponential function y = 3(2)x + 4 represents the growth of a certain species of animal in a specific area. The y-axis represents the number of that animal present and the x-axis represents years. What can you say about the growth of the population of the species every year?
A) The growth rate doubles each year.
B) The growth rate triples each year.
C) The growth rate increases by 2 each year.
D) The growth rate increases by 5 each year.
8)
Write the explicit formula for the geometric sequence represented by the recursive formula.
a1 = 11
an = 3 · an-1
A) an = 11 · 3^n
B) an = 3 · 11^n
C) an = 11 · 3^n-1
D) an = 3 · 11^n-1
10)
Describe the end behavior of f(x) =
1x/3 - 5. [Note: This is an exponential function where
1/3 is the base and x is the exponent.]
A) As x → -∞, f(x) → -5. As x → ∞, f(x) → ∞.
B) As x → -∞, f(x) → 0. As x → ∞, f(x) → ∞.
C) As x → -∞, f(x) → ∞. As x → ∞, f(x) → -5.
D) As x → -∞, f(x) → ∞. As x → ∞, f(x) → 0.
11)
Simplify the following expression
4x0 y-2 z3/4x
A)
0/4x
B)
z^3/xy^2
C)
4z^3/xy^2
D) -y^2z3
12)
Which equation represents the graph shown?
A) y = 2(x - 1) - 2
B) y = 2(x - 1) + 2
C) y = 2(x + 1) - 2
D) y = 2(x + 1) + 2
13)
Which exponential function/geometric sequence matches the graph?
A) y = -1/2^x
B) y = 1/2^x
C) y = -2x
D) y = 2x
14)
Write an equation that models the sequence 400, 200, 100, 50, ...
A) y = 400(2^x)
B) y = 50 (2^x)
C) y = 1/2x + 400
D) y = 400(1/2)^x-1
1) To find the factor by which the exponential function grows between x = 5 and x = 7, we need to calculate the value of the function at x = 7 and divide it by the value of the function at x = 5:
f(7) = 3(5)^7
f(5) = 3(5)^5
The factor of growth is therefore f(7)/f(5) = (3(5)^7) / (3(5)^5) = 25.
So, the answer is B) 25.
3) When a city triples in size every 8 years, the population growth is modeled by an exponential function. To find the population in 24 years, we need to calculate the value of the function at x = 24:
P(24) = 1000(3)^3
P(24) = 1000(27)
P(24) = 27,000
So, the population will be 27,000 in 24 years. The population growth is modeled by an exponential function.
The answer is D) 27,000; exponential.
4) In the function f(x) = 75(2)^x, the parameters represent the following:
A) The 75 is the initial number of cells, and the 2 indicates that the number of cells doubles every minute.
So, the answer is A) The 75 is the initial number of cells, and the 2 indicates that the number of cells doubles every minute.
5) The formula for the growth of the bacteria's population is A = P0 · 1.02^t.
If we begin with 200 bacteria, we can plug in the values into the formula:
A = 200 · 1.02^100/5
Simplifying the expression, we get:
A = 200 · 1.02^20
Using a calculator or computer program, we can find that 1.02^20 is approximately 1.485947.
Plugging this value back into the formula:
A = 200 · 1.485947
A ≈ 297.19
So, there will be about 297 bacteria after 100 hours.
The answer is B) 297.
6) When f(x) is replaced with f(x - 3/5), the graph of the function f(x) = 2^x is horizontally translated 3/5 units to the right.
So, the answer is D) translate horizontally 3/5 units right.
7) In the function y = 3(2)^x + 4, the growth rate of the population of the species every year is that the growth rate triples each year.
So, the answer is B) The growth rate triples each year.
8) The explicit formula for the geometric sequence represented by the recursive formula an = 3 · an-1 is:
an = 11 · 3^(n - 1)
So, the answer is C) an = 11 · 3^n-1.
10) The end behavior of f(x) = 1x/3 - 5 is as follows:
As x approaches negative infinity, f(x) approaches -5. As x approaches positive infinity, f(x) approaches infinity.
So, the answer is A) As x → -∞, f(x) → -5. As x → ∞, f(x) → ∞.
11) Simplifying the expression 4x^0 y^-2 z^3/4x, we get:
4*1*y^-2*z^3 / (4*x)
Simplifying further:
1*y^-2*z^3 / x
So, the answer is B) z^3/xy^2.
12) The equation that represents the graph shown is:
y = 2(x - 1) + 2
So, the answer is B) y = 2(x - 1) + 2.
13) From the given options, none of the exponential functions/geometric sequences provided match the given graph. The correct answer is not provided.
14) To model the given sequence 400, 200, 100, 50, ..., we can see that each term is obtained by dividing the previous term by 2. This is equivalent to multiplying by (1/2).
Therefore, the equation that models the sequence is:
y = 400 * (1/2)^x-1
So, the answer is D) y = 400(1/2)^x-1.
1) In order to determine the factor by which the exponential function grows between x = 5 and x = 7, we need to evaluate the function at these values and compare the results.
To find the value of the function at x = 5, we substitute x = 5 into the function:
f(5) = 3(5)^5 = 3(3125) = 9375
Similarly, to find the value of the function at x = 7, we substitute x = 7 into the function:
f(7) = 3(5)^7 = 3(78125) = 234375
The factor by which the function grows between x = 5 and x = 7 is the ratio of the function values at these points:
234375 / 9375 = 25
Therefore, the correct answer is B) 25.
2) The population of the city triples every 8 years. To find the population after 24 years, we need to determine how many times the population triples.
Since 24 is divisible evenly by 8, we can write it as 24 = 3(8). This means that the population triples three times in 24 years.
Starting with a population of 1000, we triple the population three times:
1000 * 3 * 3 * 3 = 1000 * 27 = 27000
Therefore, the population will be 27000 in 24 years. The population growth in this scenario is modeled by an exponential function.
The correct answer is D) 27000; exponential.
3) The exponential function f(x) = 75(2)^x represents cell duplication where the cells double every minute.
The parameter 75 represents the initial number of cells. This means that at x = 0 (the starting point), there are 75 cells.
The parameter 2 represents the factor by which the number of cells doubles every minute. Each time the value of x increases by 1, the number of cells doubles. Therefore, the number of cells at x = 1 is 75 * 2, the number of cells at x = 2 is 75 * 2 * 2, and so on.
Therefore, the correct answer is A) The 75 is the initial number of cells, and the 2 indicates that the number of cells doubles every minute.
4) To estimate the number of bacteria after 100 hours using the formula A = P0 · 1.02^t, we need to substitute the given values into the formula.
P0 represents the initial population, which is given as 200 bacteria.
t represents the time, which is 100 hours in this case.
Plugging these values into the formula, we get:
A = 200 · 1.02^100
Calculating this expression gives us approximately 297.18.
Therefore, the correct answer is B) 297.
A homework dump? Booo! Still, I'll try to get you started on these, just 'cause I'm feeling generous right now.
#1. as x grows by 2, 3*5^x grows by a factor of 5^2
#3. 24 = 8*3, so it triples 3 times, right?
#4. initial value is when x=0. 2^0 = 1...
#5. just plug in t=100
#6. f(x-h) shifts the graph to the right by h
#7. none of the above. That pesky +4 messes up the ratios
#8. well, you keep multiplying by 3 for each successive term. What does this whole set of exercises deal with?
#10. (1/3)^x gets smaller and smaller, since you keep multiplying by a number less than 1. It is like the left side of the graph of 3^x
#11. 4x^0 y^-2 z^3/4x = (4*1*z^3)/(y^2*4x) = ...
#12,13. no idea -- no graph
#14. well, you keep multiplying by 1/2, right?