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Match Exponential Growth and Decay Functions to Their Graphs
Mathematics
Grade 9 (Junior High School)
Question Content
Match the exponential function with its graph: 18. $y = 7(0.4)^t$; 19. $y = 3(1.5)^t$; 20. $y = 8(\\frac{5}{6})^t$. Graph A: decreasing curve starting high on left, approaching x-axis on right; Graph B: increasing curve starting low on left, rising on right; Graph C: decreasing curve starting high on left, approaching x-axis on right (less steep than A)
Correct Answer
18. Graph A; 19. Graph B; 20. Graph C
Detailed Solution Steps
1
Step 1: Recall the form of exponential functions: $y = ab^t$, where $a$ is the initial value (y-intercept when $t=0$) and $b$ is the growth/decay factor.
2
Step 2: For $y = 7(0.4)^t$: $b=0.4<1$, so it is a decay function. The y-intercept is $7$. It matches Graph A, which has a y-intercept near 7 and decreases steeply.
3
Step 3: For $y = 3(1.5)^t$: $b=1.5>1$, so it is a growth function. The y-intercept is $3$. It matches Graph B, which has a y-intercept near 3 and increases as t increases.
4
Step 4: For $y = 8(\\frac{5}{6})^t$: $\\frac{5}{6}≈0.83<1$, so it is a decay function. The y-intercept is $8$. It matches Graph C, which has a y-intercept near 8 and decreases gently (less steep than A, since 0.83 is closer to 1 than 0.4).
Knowledge Points Involved
1
Exponential Growth and Decay Functions
Exponential functions follow the form $y=ab^t$. When $b>1$, the function represents exponential growth (values increase as $t$ increases). When $0<b<1$, the function represents exponential decay (values decrease as $t$ increases). $a$ is the initial value, the value of $y$ when $t=0$.
2
Y-intercept of Exponential Functions
For the exponential function $y=ab^t$, substituting $t=0$ gives $y=a(1)=a$, so the y-intercept is always equal to the coefficient $a$. This is used to identify and match functions to their graphs.
3
Rate of Exponential Decay
For decay functions ($0<b<1$), the closer $b$ is to 1, the slower the decay (less steep the curve). The closer $b$ is to 0, the faster the decay (steeper the curve).
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