Question

Linear and Nonlinear Functions
x(years) F(x) G(x)
0 50 20
1 75 30
2 100 45
3 125 67.5
4 150 101.25
5 175 157.88
6 200 227.82
Question 1
Frank purchases a 1920 penny worth $50, and George purchases a 1980 penny valued at $20. The table shows the value of Frank's penny, F(x), and the value of George's penny, G(x) after x years. Which statement is TRUE about F(x) and G(x)?
Responses
A F(x) is an exponential function, and G(x) is a linear function.F(x) is an exponential function, and G(x) is a linear function.
B F(x) is a linear function, and G(x) is an exponential function.F(x) is a linear function, and G(x) is an exponential function.
C Both F(x) and G(x) are exponential functions.Both F(x) and G(x) are exponential functions.
D Both F(x) and G(x) are linear functions.Both F(x) and G(x) are linear functions.
Question 2
If the value of each penny continues as shown, how much more will George's penny be worth than Frank's at the end of year 7? (nearest dollar)
Responses
A $340$340
B $117$117
C $91$91
D $76

Answer 1

To analyze the functions F(x) and G(x) and address both questions, we start by examining the values given for each function up to year 6.

### Question 1
We need to determine whether F(x) and G(x) are linear or exponential functions.

1. A **linear function** will show a constant rate of change (constant difference between consecutive y-values).
2. An **exponential function** will show a constant percentage change, where each y-value grows by a constant ratio.

Examining F(x):
- Differences between consecutive F(x) values:
- 75 - 50 = 25
- 100 - 75 = 25
- 125 - 100 = 25
- 150 - 125 = 25
- 175 - 150 = 25
- 200 - 175 = 25
- The increases are constant, indicating that F(x) is a linear function.

Examining G(x):
- Differences between consecutive G(x) values:
- 30 - 20 = 10
- 45 - 30 = 15
- 67.5 - 45 = 22.5
- 101.25 - 67.5 = 33.75
- 157.88 - 101.25 = 56.63
- 227.82 - 157.88 = 69.94
- The increases are not constant and are increasing, indicating that G(x) is an exponential function.

Thus, the true statement is:
**A:** F(x) is an exponential function, and G(x) is a linear function. (This statement seems incorrect based on our analysis, so we need to revisit the answer based on simpler interpretations.)

However, based on conventional interpretations:
**B:** F(x) is a linear function, and G(x) is an exponential function.

### Question 2
Now let's calculate the value of both pennies at the end of year 7 using the previously established rates.

For F(x) (a linear function):
- The slope is 25 (derived from previous calculations).
- At year 7: F(7) = F(6) + 25 = 200 + 25 = 225

For G(x) (an exponential function):
To find G(7), we examine the ratio of changes to detect multiplication, but for brevity:
Calculating G(7):
- We see that each increase is growing approximatively as a geometric series, to first approximations based on the last three computations, we can see the pattern seems to echo closest to 1.3 multiples.

Hence let's add to the last recorded:
- At 6: G(6) = 227.82
- We will cautiously assume next year would multiply somewhat by a factor in similar approximative sequences (~factor of 1.3).

Let's multiply directly just for year 7:
\[ G(7) \approx 1.3 \cdot 227.82 \approx 295.13 (if we stretch based off observed) \]

Finally, the difference:
- \( G(7) - F(7) = 295.13 - 225 = 70.13 (nearest dollar 70)\)

Given the options:
**D:** $76 is the best fit, as rounding lower within the option adherences.

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Thus, to summarize:
1. F(x) is linear, G(x) is exponential: **B**.
2. Difference at year 7: **D - $76**.