Exponential Graphs and Equations Free Worksheet | Fun and Engaging ... - Free Printable

Problem Overview:

The task involves matching exponential equations to their corresponding graphs. There are two sections:

1. Section A: Match the graph to the exponential equation \( y = a^x \) or \( y = a^{-x} \).
2. Section B: Substitute \( x = 0 \) into the given equations to help label the graphs.

Let's solve each section step by step.

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Section A: Matching Graphs to Equations

#### Given Equations:
- \( A \): \( y = 6^x \)
- \( B \): \( y = 1.5^{-x} \)
- \( C \): \( y = 0.2^x \)
- \( D \): \( y = 2.5^x \)

#### Key Observations:
1. Exponential Growth vs. Decay:
- If \( a > 1 \), the function \( y = a^x \) represents exponential growth (increasing as \( x \) increases).
- If \( 0 < a < 1 \), the function \( y = a^x \) represents exponential decay (decreasing as \( x \) increases).

2. Negative Exponent:
- \( y = a^{-x} \) is equivalent to \( y = \left(\frac{1}{a}\right)^x \). This means:
- If \( a > 1 \), \( y = a^{-x} \) will behave like \( y = b^x \) where \( 0 < b < 1 \) (exponential decay).
- If \( 0 < a < 1 \), \( y = a^{-x} \) will behave like \( y = b^x \) where \( b > 1 \) (exponential growth).

#### Analyzing the Graph:
- The graph shows four curves:
- Two curves are increasing (exponential growth).
- Two curves are decreasing (exponential decay).

#### Matching the Curves:
1. Increasing Curves:
- These correspond to equations where \( a > 1 \):
- \( y = 6^x \) (steepest growth, labeled as \( A \)).
- \( y = 2.5^x \) (slower growth, labeled as \( D \)).

2. Decreasing Curves:
- These correspond to equations where \( 0 < a < 1 \) or \( a^{-x} \):
- \( y = 0.2^x \) (rapid decay, labeled as \( C \)).
- \( y = 1.5^{-x} \) (slower decay, labeled as \( B \)).

#### Final Labels:
- Red Curve: \( y = 6^x \) (steepest growth) → A
- Blue Curve: \( y = 2.5^x \) (slower growth) → D
- Green Curve: \( y = 0.2^x \) (rapid decay) → C
- Orange Curve: \( y = 1.5^{-x} \) (slower decay) → B

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Section B: Labeling Graphs Using \( x = 0 \)

#### Given Equations:
1. \( y = 3(2^x) \)
2. \( y = 0.5(5^{-x}) \)
3. \( y = 2(3^x) \)
4. \( y = 2(4^{-x}) \)
5. \( y = 4(0.5^x) \)

#### Strategy:
- Substitute \( x = 0 \) into each equation to find the \( y \)-intercept.
- Use the \( y \)-intercepts to match the graphs.

#### Calculations:
1. \( y = 3(2^x) \):
\[
y = 3(2^0) = 3(1) = 3
\]
- \( y \)-intercept: 3

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

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

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

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

#### Observations:
- The \( y \)-intercepts are:
- 3, 0.5, 2, 2, 4.
- From the graph, the \( y \)-intercepts of the curves are:
- Blue curve: 4
- Red curve: 3
- Green curve: 2
- Orange curve: 2
- Purple curve: 0.5

#### Matching:
- Blue Curve: \( y = 4(0.5^x) \) → 5
- Red Curve: \( y = 3(2^x) \) → 1
- Green Curve: \( y = 2(3^x) \) → 3
- Orange Curve: \( y = 2(4^{-x}) \) → 4
- Purple Curve: \( y = 0.5(5^{-x}) \) → 2

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Final Answers:

#### Section A:
- Red Curve: \( \boxed{A} \)
- Blue Curve: \( \boxed{D} \)
- Green Curve: \( \boxed{C} \)
- Orange Curve: \( \boxed{B} \)

#### Section B:
- Blue Curve: \( \boxed{5} \)
- Red Curve: \( \boxed{1} \)
- Green Curve: \( \boxed{3} \)
- Orange Curve: \( \boxed{4} \)
- Purple Curve: \( \boxed{2} \)