Constructing Exponential Functions (Grades 11-12)

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Constructing Exponential Functions

1. 
What is the equation of the exponential function graphed below? Assume that the scale of both axes is one unit.
Graph - Exponent Function y=2^x
  1. [math]f(x) = 2^x[/math]
  2. [math]f(x) = 10^x[/math]
  3. [math]f(x) = e^x[/math]
  4. Not enough information
2. 
What is the equation of the exponential function in the following graph? Assume that the scale of both axes is one unit.
Graph - Exponent Function y=5^x
  1. [math]f(x) = 2^x[/math]
  2. [math]f(x) = 2^x + 3[/math]
  3. [math]f(x) = 5^x[/math]
  4. [math]f(x) = 10^x - 5[/math]
3. 
The following table gives some of the values of [math]f(x)[/math]. Using these values, what is the function rule for this exponential function?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{x} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{f(x)} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 0 [/math][math] 5 [/math]
[math] 1 [/math][math] 10 [/math]
[math] 2 [/math][math] 20 [/math]
[math] 3 [/math][math] 40 [/math]
  1. [math]f(x) = 2*5^x[/math]
  2. [math]f(x) = 5*e^x[/math]
  3. [math]f(x) = 2^x + 5[/math]
  4. [math]f(x) = 5*2^x[/math]
4. 
Create an exponential function of the form [math]f(x) = ab^x,[/math] given the points [math](-2,5)[/math] and [math](3,10)[/math]. The numerical values of [math]a[/math] and [math]b[/math] have been rounded in the following answers.
  1. [math]f(x) = 6.61*1.15^x[/math]
  2. [math]f(x) = 0.10*0.14^x[/math]
  3. [math]f(x) = 7.94*1.26^x[/math]
  4. At least three points are needed.
5. 
Given the points [math](1,2)[/math] and [math](8,9)[/math], which of the following exponential functions could be created with them? The numerical values in the function have been rounded.
  1. [math]f(x) = 1.41*1.42^x[/math]
  2. [math]f(x) = 1.61*1.24^x[/math]
  3. [math]f(x) = 2.74*1.16^x[/math]
  4. All of the above.
6. 
There are 20 rabbits living on an island, and the population triples every year. Which of the following functions correctly models this situation, where [math]P[/math] is the population of rabbits, and [math]t[/math] is the number of years since the current population count?
  1. [math]P(t) = 20 + 3^t[/math]
  2. [math]P(t) = 20t^3[/math]
  3. [math]P(t) = 3t + 2[/math]
  4. [math]P(t) = 20*3^t[/math]
7. 
There is a 100 g sample of Sodium-24 sitting in a secure storage box. After 15 hours, there is only 50 g left. Which function best describes this relationship between the amount of Sodium-24, [math]S[/math], and the time that has elapsed, in hours, [math]t[/math]?
  1. [math]S(t) = 100*(1/2)^(15t)[/math]
  2. [math]S(t) = 100 e^(-0.046t)[/math]
  3. [math]S(t) = 50*(1/2)^(15t)[/math]
  4. [math]S(t) = 50 e^(-0.5t)[/math]
8. 
A lab keeps blood samples in a special fridge, which is kept at 0 °C, for a certain experiment. After the blood is initially taken from a patient and put in the fridge, its temperature is checked every 30 minutes for the first two hours. Below are those recordings. Which exponential function rule best describes this information?

[math] \ \ \ \ \ \ \ \ \ \ \ mathbf{"Time in Fridge (min)"} \ \ \ \ \ \ \ \ \ \ \ [/math][math] \ \ \ \ \ \ \ \ \ \ \mathbf{"Temperature (°C)"} \ \ \ \ \ \ \ \ \ \ [/math]
[math] 30 [/math][math] 19.2 [/math]
[math] 60 [/math][math] 10.5 [/math]
[math] 90 [/math][math] 5.8 [/math]
[math] 120 [/math][math] 3.2 [/math]
  1. [math]f(t) = 35.2 * (-0.02)^t[/math]
  2. [math]f(t) = 35.2 * 0.98^t [/math]
  3. [math]f(t) = 6.8 * 1.08^t[/math]
  4. There is not enough information given in the table to find the function rule.
9. 
A store relies on word of mouth advertising so it brings in 30 potential customers and gives them $100 gift cards if they will each tell 5 friends within a week. If each of those people tell 5 more friends within a week and the pattern continues, how many people will hear about the store after 10 weeks?
  1. 488,281,250
  2. 624,431,250
  3. 292,968,750
  4. 128,281,250
10. 
A bacteria colony covers [math]5 \ "cm"^2[/math] and grows exponentially. After 4 days it covers [math]80 \ "cm"^2[/math]. How much area will it cover after 2 weeks?
  1. [math]8.2 \ "m"^2[/math]
  2. [math]4300 \ "cm"^2[/math]
  3. [math]4.3 \ "m"^2[/math]
  4. [math]1393 \ "cm"^2[/math]

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