Question

model & discuss
a radio station uses the function $f(x) = 100(3)^x$ to model the growth of band a’s fan base.
image of a graph titled band as fan base with x-axis time (months) and y-axis number of fans, showing the curve of $f(x) = 100(3)^x$
a. what would the graph of the function look like for band b with a fan base growing twice as fast as band a’s fan base?
b. compare and contrast the two graphs.
c. look for relationships suppose band c starts with a fan base of 200 fans that is growing twice as fast as band a’s fan base. compare and contrast this new function with the previous two functions.

Explanation:

Part A

Step1: Analyze Band A's function

Band A's fan base growth is modeled by \( f(x)=100(3)^{x} \), where the base of the exponential function is 3 (representing the growth factor per unit time \( x \) (months)).

Step2: Determine Band B's growth factor

If Band B's fan base grows twice as fast as Band A's, the growth factor of Band B will be \( 3\times2 = 6 \)? Wait, no. Wait, the "rate" in exponential growth: the general form is \( f(x)=a(b)^{x} \), where \( b = 1 + r \), \( r \) is the growth rate. If the growth rate of B is twice that of A, then the growth factor \( b \) for B should be such that the rate \( r_B = 2r_A \). For Band A, \( b_A=3 \), so \( r_A=3 - 1=2 \) (wait, no, that's not the right way. Wait, actually, the function is \( f(x)=a(\text{growth factor})^x \). If the growth "speed" is twice as fast, the growth factor should be \( 3^2 \)? No, wait, no. Wait, let's think again. The function for Band A is \( f(x)=100(3)^x \). The exponent is \( x \), so the growth is exponential with base 3. If Band B grows twice as fast, that means for each unit of time \( x \), the multiplier is squared? Wait, no. Wait, the rate of growth: the derivative (rate of change) of \( f(x)=a(b)^x \) is \( f^\prime(x)=a(b)^x\ln(b) \). So if the rate of B is twice that of A, then \( a_B(b_B)^x\ln(b_B)=2\times a_A(b_A)^x\ln(b_A) \). But since \( a_B = a_A = 100 \) (assuming same initial amount? Wait, the problem says "fan base growing twice as fast" – maybe it's the growth factor that is squared? Wait, no, maybe the base of the exponent is multiplied by 2? Wait, no, let's re - read the problem: "fan base growing twice as fast as Band A’s fan base". The function for Band A is \( f(x)=100(3)^x \). So the growth factor per month is 3. If Band B grows twice as fast, the growth factor per month should be \( 3\times2 = 6 \)? Wait, no, that would be if the growth rate (multiplier) is doubled. Wait, actually, the correct way: the general form of exponential growth is \( y = a(1 + r)^x \), where \( r \) is the growth rate. For Band A, \( 1 + r_A=3 \), so \( r_A = 2 \). Then for Band B, \( r_B = 2r_A=4 \), so \( 1 + r_B=5 \)? No, this is confusing. Wait, maybe the problem means that the base of the exponential function is squared? Wait, no, let's think of the function. If the growth is "twice as fast", the function for Band B should be \( f_B(x)=100(3^{2})^x=100(9)^x \)? Wait, no, that would be if the growth factor is squared. Wait, no, the exponent rules: \( (a^m)^n=a^{mn} \). Wait, maybe the problem is simpler: if the growth is twice as fast, the base of the exponential function (the growth factor) is multiplied by 2. So Band A has \( f(x)=100(3)^x \), Band B, growing twice as fast, would have \( f(x)=100(3\times2)^x=100(6)^x \)? Wait, but maybe the intended interpretation is that the growth factor is squared, i.e., \( 3^2 = 9 \), so \( f_B(x)=100(9)^x \). Wait, let's check with \( x = 1 \). For Band A, \( f(1)=100\times3 = 300 \). If Band B grows twice as fast, at \( x = 1 \), \( f_B(1) \) should be \( 300\times2=600 \). Let's see: if \( f_B(x)=100(6)^x \), then \( f_B(1)=100\times6 = 600 \), which matches. If \( f_B(x)=100(9)^x \), then \( f_B(1)=900 \), which is three times as much. So the correct growth factor is 6? Wait, no, the growth rate: the growth factor is \( b \), so the amount at time \( x + 1 \) is \( b \) times the amount at time \( x \). So if Band B grows twice as fast as Band A, then \( \frac{f_B(x + 1)}{f_B(x)}=2\times\frac{f_A(x + 1)}{f_A(x)} \). Since \( \frac{f_A(x + 1)}{f_A(x)} = 3 \), then \( \frac{f_B(x + 1)}{f_B(x)}=2\times3 =…

Step1: Compare the y - intercepts

For Band A's function \( f_A(x)=100(3)^x \) and Band B's function \( f_B(x)=100(6)^x \), when \( x = 0 \), \( f_A(0)=100(3)^0=100 \) and \( f_B(0)=100(6)^0 = 100 \). So both graphs pass through the point (0, 100), meaning they have the same initial fan base (y - intercept).

Step2: Compare the growth rates (steepness)

The growth factor of Band A is 3, and the growth factor of Band B is 6. Since 6>3, the graph of Band B's function \( f_B(x)=100(6)^x \) is steeper than the graph of Band A's function \( f_A(x)=100(3)^x \). This means that for \( x>0 \), the number of fans for Band B increases at a faster rate than for Band A, so the values of \( f_B(x) \) will be greater than \( f_A(x) \) for \( x>0 \).

Step3: Compare the general shape

Both graphs are exponential growth curves, which means they are increasing functions, and their graphs are concave up (curving upwards as \( x \) increases).

Step1: Determine Band C's function

Band C starts with a fan base of 200 fans, so \( a = 200 \). It grows twice as fast as Band A, so the growth factor is the same as Band B's growth factor, which is 6 (since the growth rate is twice that of Band A, and we found earlier that the growth factor for twice the rate of Band A is 6). So the function for Band C is \( f_C(x)=200(6)^x \).

Step2: Compare with Band A's function

• Y - intercept: Band A has a y - intercept of (0, 100), Band C has a y - intercept of (0, 200) (since \( 200(6)^0=200 \)).
• Growth factor: Band A has a growth factor of 3, Band C has a growth factor of 6. So Band C's graph is steeper than Band A's.
• Initial amount: Band C starts with more fans (200 vs. 100) and grows faster.

Step3: Compare with Band B's function

• Y - intercept: Band B has a y - intercept of (0, 100), Band C has a y - intercept of (0, 200).
• Growth factor: Both Band B and Band C have a growth factor of 6, so their graphs have the same steepness (same rate of growth in terms of the multiplier per unit time).
• Initial amount: Band C starts with more fans (200 vs. 100), so at \( x = 0 \), Band C has more fans, and for \( x>0 \), since they have the same growth factor, the difference in the number of fans between Band C and Band B will remain proportional to their initial amounts (Band C will always have twice as many fans as Band B at the same time \( x \), because \( \frac{f_C(x)}{f_B(x)}=\frac{200(6)^x}{100(6)^x}=2 \)).

Answer:

(Part A):
The function for Band B is \( f(x)=100(6)^{x} \), and its graph will be an exponential growth curve with the same y - intercept (0, 100) as Band A's graph but steeper (rising more rapidly) because it has a larger growth factor (6 vs. 3).

Part B