From the given graph and the equations, we need to match each equation to the corresponding graph based on the shape and growth/decay pattern. Here's the reasoning:

Key Points to Match the Graph:

1. Exponential Growth:

   • If $b > 1$ in $y = a(b)^z$, the graph represents exponential growth.
   • Larger values of $b > 1$ will grow faster (steeper upward curves).

2. Exponential Decay:

   • If $0 < b < 1$, the graph represents exponential decay.
   • Smaller values of $b$ (closer to 0) will decay faster.

3. Intercept and Stretch (Initial Value):

   • The value of $a$ affects the y-intercept and stretch of the graph.

Analyzing Each Equation and Graph:

1. $y = 2(1.18)^z$:

   • Exponential growth ($b = 1.18 > 1$) and slower than others with $b$ close to 1.
   • Matches green (G).

2. $y = 3(0.68)^z$:

   • Exponential decay ($b = 0.68 < 1$) and decays faster than $0.84$.
   • Matches black (K).

3. $y = 3(1.18)^z$:

   • Exponential growth ($b = 1.18 > 1$) and faster growth due to larger $a$.
   • Matches red (R).

4. $y = 3(0.84)^z$:

   • Exponential decay ($b = 0.84 < 1$) and decays slower than $0.68$.
   • Matches blue (B).

5. $y = 3(1.47)^z$:

   • Exponential growth ($b = 1.47 > 1$) and very steep growth.
   • Matches orange (O).

Final Match:

• $y = 2(1.18)^z$: green (G)
• $y = 3(0.68)^z$: black (K)
• $y = 3(1.18)^z$: red (R)
• $y = 3(0.84)^z$: blue (B)
• $y = 3(1.47)^z$: orange (O)

Would you like a more detailed explanation or graph sketch?

Related Questions:

1. What is the significance of the base $b$ in an exponential function?
2. How does the value of $a$ affect the graph of $y = a(b)^z$?
3. Why does $b > 1$ result in growth and $0 < b < 1$ result in decay?
4. How can we visually compare exponential growth rates from graphs?
5. How can this concept of exponential functions apply to real-world scenarios like population growth or radioactive decay?

Tip:

When matching equations to graphs, focus on both the growth/decay pattern (base $b$) and the initial value (constant $a$).