Exponential Functions worksheet with graphs and equations for matching and substitution practice.

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 \( x \) increases, \( y \) grows rapidly. The graph will start near 1 when \( x = 0 \) and increase steeply.
- \( y = 1.5^{-x} \): 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 because the base \( \frac{2}{3} \) is between 0 and 1. As \( x \) increases, \( y \) decreases toward 0.
- \( y = 0.2^x \): This is an exponential decay function because the base 0.2 is between 0 and 1. As \( x \) increases, \( y \) decreases toward 0.
- \( y = 2.5^x \): This is an exponential growth function with a base greater than 1. As \( x \) increases, \( y \) grows rapidly. The graph will start near 1 when \( x = 0 \) and increase steeply, but not as steeply as \( y = 6^x \).

##### Step 2: Match the equations to the graphs
- Graph \( A \): This graph shows rapid exponential growth. It matches \( y = 6^x \).
- Graph \( B \): This graph shows exponential decay. It matches \( y = 0.2^x \).
- Graph \( C \): This graph shows slower exponential growth. It matches \( y = 2.5^x \).
- Graph \( D \): This graph shows exponential decay. It matches \( y = 1.5^{-x} \).

##### Final Answer for Section A
\[
\boxed{A = y = 6^x, \, B = y = 0.2^x, \, C = y = 2.5^x, \, D = y = 1.5^{-x}}
\]

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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, and 5.

##### Step 1: Evaluate each function at \( x = 0 \)
- Equation 1: \( y = 3(2^x) \)
\[
y = 3(2^0) = 3(1) = 3
\]
When \( x = 0 \), \( y = 3 \).

- Equation 2: \( y = 0.5(5^{-x}) \)
\[
y = 0.5(5^0) = 0.5(1) = 0.5
\]
When \( x = 0 \), \( y = 0.5 \).

- Equation 3: \( y = 2(3^x) \)
\[
y = 2(3^0) = 2(1) = 2
\]
When \( x = 0 \), \( y = 2 \).

- Equation 4: \( y = 2(4^{-x}) \)
\[
y = 2(4^0) = 2(1) = 2
\]
When \( x = 0 \), \( y = 2 \).

- Equation 5: \( y = 4(0.5^x) \)
\[
y = 4(0.5^0) = 4(1) = 4
\]
When \( x = 0 \), \( y = 4 \).

##### Step 2: Analyze the behavior of each function
- Equation 1: \( y = 3(2^x) \): This is an exponential growth function. As \( x \) increases, \( y \) grows rapidly.
- Equation 2: \( y = 0.5(5^{-x}) \): This is an exponential growth function because \( 5^{-x} = \left( \frac{1}{5} \right)^{-x} = 5^x \). As \( x \) increases, \( y \) grows.
- Equation 3: \( y = 2(3^x) \): This is an exponential growth function. As \( x \) increases, \( y \) grows rapidly.
- Equation 4: \( y = 2(4^{-x}) \): This is an exponential decay function because \( 4^{-x} = \left( \frac{1}{4} \right)^x \). As \( x \) increases, \( y \) decreases toward 0.
- Equation 5: \( y = 4(0.5^x) \): This is an exponential decay function. As \( x \) increases, \( y \) decreases toward 0.

##### Step 3: Match the equations to the graphs using the values at \( x = 0 \) and behavior
- Graph 1: Starts at \( y = 4 \) and decays. Matches \( y = 4(0.5^x) \).
- Graph 2: Starts at \( y = 2 \) and decays. Matches \( y = 2(4^{-x}) \).
- Graph 3: Starts at \( y = 3 \) and grows. Matches \( y = 3(2^x) \).
- Graph 4: Starts at \( y = 2 \) and grows. Matches \( y = 2(3^x) \).
- Graph 5: Starts at \( y = 0.5 \) and grows. Matches \( y = 0.5(5^{-x}) \).

##### Final Answer for Section B
\[
\boxed{1 = y = 4(0.5^x), \, 2 = y = 2(4^{-x}), \, 3 = y = 3(2^x), \, 4 = y = 2(3^x), \, 5 = y = 0.5(5^{-x})}
\]

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Final Answers

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