This worksheet guides students through matching exponential equations to their corresponding graphs and identifying y-intercepts.

Problem Analysis and Solution

The task involves matching exponential functions to their corresponding graphs. We will solve this step by step for both Section A and Section B.

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#### Section A: Match the graph to the exponential equation

We are given four equations:
1. \( y = 6^x \)
2. \( y = 1.5^{-x} \)
3. \( y = 0.2^x \)
4. \( y = 2.5^x \)

And we need to match these equations to the graphs labeled as \( A, B, C, \) and \( D \).

##### Step 1: Analyze the behavior of each function
- \( y = 6^x \): This is an exponential growth function with a base greater than 1. As \( x \to \infty \), \( y \to \infty \), and as \( x \to -\infty \), \( y \to 0 \). The graph will be increasing.
- \( y = 1.5^{-x} \): This can be rewritten as \( y = \left( \frac{1}{1.5} \right)^x = \left( \frac{2}{3} \right)^x \). This is an exponential decay function with a base between 0 and 1. As \( x \to \infty \), \( y \to 0 \), and as \( x \to -\infty \), \( y \to \infty \). The graph will be decreasing.
- \( y = 0.2^x \): This is an exponential decay function with a base between 0 and 1. As \( x \to \infty \), \( y \to 0 \), and as \( x \to -\infty \), \( y \to \infty \). The graph will be decreasing.
- \( y = 2.5^x \): This is an exponential growth function with a base greater than 1. As \( x \to \infty \), \( y \to \infty \), and as \( x \to -\infty \), \( y \to 0 \). The graph will be increasing.

##### Step 2: Match the graphs
- Graph \( A \): This graph shows exponential growth (increasing). It corresponds to \( y = 6^x \) because 6 is the largest base among the growth functions.
- Graph \( B \): This graph shows exponential decay (decreasing). It corresponds to \( y = 0.2^x \) because 0.2 is the smallest base among the decay functions.
- Graph \( C \): This graph shows exponential decay (decreasing). It corresponds to \( y = 1.5^{-x} \) because \( \frac{2}{3} \) is slightly larger than 0.2.
- Graph \( D \): This graph shows exponential growth (increasing). It corresponds to \( y = 2.5^x \) because 2.5 is smaller than 6 but still a growth function.

##### Final Matching for Section A
- \( A \): \( y = 6^x \)
- \( B \): \( y = 0.2^x \)
- \( C \): \( y = 1.5^{-x} \)
- \( D \): \( y = 2.5^x \)

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#### Section B: Substitute \( x = 0 \) into the equations to help label the graphs

We are given five equations:
1. \( y = 3(2^x) \)
2. \( y = 0.5(5^{-x}) \)
3. \( y = 2(3^x) \)
4. \( y = 2(4^{-x}) \)
5. \( y = 4(0.5^x) \)

And we need to match these equations to the graphs labeled as 1, 2, 3, 4, or 5.

##### Step 1: Evaluate each function at \( x = 0 \)
- Equation 1: \( y = 3(2^x) \)
\[
y(0) = 3(2^0) = 3(1) = 3
\]
- Equation 2: \( y = 0.5(5^{-x}) \)
\[
y(0) = 0.5(5^0) = 0.5(1) = 0.5
\]
- Equation 3: \( y = 2(3^x) \)
\[
y(0) = 2(3^0) = 2(1) = 2
\]
- Equation 4: \( y = 2(4^{-x}) \)
\[
y(0) = 2(4^0) = 2(1) = 2
\]
- Equation 5: \( y = 4(0.5^x) \)
\[
y(0) = 4(0.5^0) = 4(1) = 4
\]

##### Step 2: Analyze the behavior of each function
- Equation 1: \( y = 3(2^x) \) — Exponential growth (base 2).
- Equation 2: \( y = 0.5(5^{-x}) \) — Exponential decay (base \( \frac{1}{5} \)).
- Equation 3: \( y = 2(3^x) \) — Exponential growth (base 3).
- Equation 4: \( y = 2(4^{-x}) \) — Exponential decay (base \( \frac{1}{4} \)).
- Equation 5: \( y = 4(0.5^x) \) — Exponential decay (base 0.5).

##### Step 3: Match the graphs using the values at \( x = 0 \) and behavior
- Graph 1: This graph shows exponential growth starting at \( y = 3 \). It corresponds to \( y = 3(2^x) \).
- Graph 2: This graph shows exponential decay starting at \( y = 0.5 \). It corresponds to \( y = 0.5(5^{-x}) \).
- Graph 3: This graph shows exponential growth starting at \( y = 2 \). It corresponds to \( y = 2(3^x) \).
- Graph 4: This graph shows exponential decay starting at \( y = 2 \). It corresponds to \( y = 2(4^{-x}) \).
- Graph 5: This graph shows exponential decay starting at \( y = 4 \). It corresponds to \( y = 4(0.5^x) \).

##### Final Matching for Section B
- 1: \( y = 3(2^x) \)
- 2: \( y = 0.5(5^{-x}) \)
- 3: \( y = 2(3^x) \)
- 4: \( y = 2(4^{-x}) \)
- 5: \( y = 4(0.5^x) \)

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Final Answer

\[
\boxed{
\text{Section A: } A = y = 6^x, B = y = 0.2^x, C = y = 1.5^{-x}, D = y = 2.5^x \\
\text{Section B: } 1 = y = 3(2^x), 2 = y = 0.5(5^{-x}), 3 = y = 2(3^x), 4 = y = 2(4^{-x}), 5 = y = 4(0.5^x)
}
\]