Question

determine which of the following exponential formula(s) represents exponential function ii in the graph above
(a) $10(1.2)^t$ \t\t\t\t\t\t\t\t($\beta$) $10(1.5)^t$
($\chi$) $20(1.2)^t$ \t\t\t\t\t\t\t\t($\delta$) $30(0.85)^t$
($\epsilon$) $30(0.95)^t$ \t\t\t\t\t\t\t\t($\theta$) $30(1.05)^t$
a. $\chi$ \t\t\t\t\t\t\t\t\t\t\t\tc. $\delta$
b. $\alpha$ \t\t\t\t\t\t\t\t\t\t\td. $\phi$

Explanation:

Step1: Analyze Exponential Growth/Decay

Exponential functions: \( y = a(b)^t \). If \( b > 1 \), growth; \( 0 < b < 1 \), decay. Function II (let's assume from graph) – check initial value (\( t=0 \): \( y = a \)) and growth/decay.

Step2: Check Initial Values

• (a): \( t=0 \), \( y=10(1.2)^0 = 10 \)
• (b): \( t=0 \), \( y=10(1.5)^0 = 10 \)
• (c): \( t=0 \), \( y=20(1.2)^0 = 20 \)
• (d): \( t=0 \), \( y=30(0.85)^0 = 30 \)
• (e): \( t=0 \), \( y=30(0.95)^0 = 30 \)
• (f): \( t=0 \), \( y=30(1.05)^0 = 30 \)

Step3: Growth/Decay from Graph

If Function II is decaying (since some curves decrease), check \( b < 1 \). (d) \( b=0.85 \), (e) \( b=0.95 \), (f) \( b=1.05 \) (growth). From initial value, if Function II has initial value 30 and decays, (d) or (e). But let's check the graph's initial point (y-intercept). Suppose the graph for II starts at 30 (y=30 when t=0) and decays. So (d) \( 30(0.85)^t \) or (e) \( 30(0.95)^t \). But the options' choices: looking at the options given (a, b, c, d, e, f) and the choices below (a, b, c, d with labels). Wait, the problem's options: (a) 10(1.2)^t, (b)10(1.5)^t, (c)20(1.2)^t, (d)30(0.85)^t, (e)30(0.95)^t, (f)30(1.05)^t. The choices below: a. z? No, maybe typo. Wait, the correct approach: exponential decay (since the curve is decreasing) has \( 0 < b < 1 \), and initial value \( a \). From the graph, if the y-intercept is 30 (since (d), (e), (f) have a=30), and it's decaying (so b < 1). So (d) \( 30(0.85)^t \) or (e) \( 30(0.95)^t \). But the options' choices: looking at the given choices (a, b, c, d with labels: a. z? No, maybe the correct option is d (30(0.85)^t) as it's a decay function with a=30.

Answer:

c. d (30(0.85)^t)