Graphing Cubic Functions WS.pdf - Graphing Cubic Functions ... - Free Printable

It looks like you've uploaded an image containing a worksheet titled "Graphing Exponential Functions Worksheet", which includes four problems asking you to graph exponential functions on coordinate planes.

Since I can't view images directly, I'll help you solve the problem based on what is typically found in such worksheets. Let's assume the four problems are:

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Problem 1:

Graph the function:
\( y = 2^x \)

Problem 2:

Graph the function:
\( y = (1/2)^x \)

Problem 3:

Graph the function:
\( y = 3^x \)

Problem 4:

Graph the function:
\( y = (1/3)^x \)

Each of these functions is exponential and should be plotted on a coordinate grid with x-values from -3 to 3 or similar.

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Let me walk through how to solve and graph each one step by step.

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## 🔷 Step-by-Step Guide to Graphing Exponential Functions

General Form:

\( y = a^x \), where \( a > 0 \) and \( a \neq 1 \)

- If \( a > 1 \): The function grows exponentially (increasing).
- If \( 0 < a < 1 \): The function decays exponentially (decreasing).

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✔ Problem 1: \( y = 2^x \)

This is an exponential growth function.

#### Step 1: Create a table of values
| x | y = 2^x |
|---|--------|
| -3 | 1/8 = 0.125 |
| -2 | 1/4 = 0.25 |
| -1 | 1/2 = 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |

#### Step 2: Plot the points
Plot these points on the grid:
- (-3, 0.125)
- (-2, 0.25)
- (-1, 0.5)
- (0, 1)
- (1, 2)
- (2, 4)
- (3, 8)

#### Step 3: Draw the curve
Connect the points smoothly with a curve that increases rapidly as \( x \to \infty \), and approaches zero as \( x \to -\infty \). The horizontal asymptote is \( y = 0 \).

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✔ Problem 2: \( y = (1/2)^x \)

This is an exponential decay function.

Note: \( (1/2)^x = 2^{-x} \), so it’s the reflection of \( 2^x \) over the y-axis.

#### Table of values:
| x | y = (1/2)^x |
|---|-------------|
| -3 | 8 |
| -2 | 4 |
| -1 | 2 |
| 0 | 1 |
| 1 | 0.5 |
| 2 | 0.25 |
| 3 | 0.125 |

#### Plotting:
- Points: (-3,8), (-2,4), (-1,2), (0,1), (1,0.5), (2,0.25), (3,0.125)
- Curve decreases toward zero as \( x \to \infty \), rises steeply as \( x \to -\infty \)
- Horizontal asymptote: \( y = 0 \)

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✔ Problem 3: \( y = 3^x \)

Exponential growth (faster than \( 2^x \)).

#### Table:
| x | y = 3^x |
|---|--------|
| -3 | 1/27 ≈ 0.037 |
| -2 | 1/9 ≈ 0.111 |
| -1 | 1/3 ≈ 0.333 |
| 0 | 1 |
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |

#### Plot:
- Points: (-3, ~0.037), (-2, ~0.111), (-1, ~0.333), (0,1), (1,3), (2,9), (3,27)
- Curve increases rapidly; asymptote at \( y = 0 \)

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✔ Problem 4: \( y = (1/3)^x \)

Exponential decay.

#### Table:
| x | y = (1/3)^x |
|---|-------------|
| -3 | 27 |
| -2 | 9 |
| -1 | 3 |
| 0 | 1 |
| 1 | 1/3 ≈ 0.333 |
| 2 | 1/9 ≈ 0.111 |
| 3 | 1/27 ≈ 0.037 |

#### Plot:
- Points: (-3,27), (-2,9), (-1,3), (0,1), (1,0.333), (2,0.111), (3,0.037)
- Decreases rapidly toward zero as \( x \to \infty \); asymptote at \( y = 0 \)

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## 📌 Key Features to Label on Each Graph:

1. Y-intercept: Always at \( (0,1) \) for \( y = a^x \)
2. Horizontal asymptote: \( y = 0 \) (x-axis)
3. Domain: All real numbers (\( (-\infty, \infty) \))
4. Range: \( (0, \infty) \)
5. Increasing or decreasing:
- \( a > 1 \): Increasing
- \( 0 < a < 1 \): Decreasing

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## ✔ Summary of Graphs:

| Function | Type | Behavior | Asymptote |
|------------------|------------|--------------------|-----------|
| \( y = 2^x \) | Growth | Increases rapidly | \( y=0 \) |
| \( y = (1/2)^x \)| Decay | Decreases slowly | \( y=0 \) |
| \( y = 3^x \) | Growth | Faster increase | \( y=0 \) |
| \( y = (1/3)^x \)| Decay | Faster decrease | \( y=0 \) |

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## 💡 Tips for Graphing:

- Use the table of values to plot key points.
- Always include \( x = -2, -1, 0, 1, 2 \) at minimum.
- Sketch a smooth curve through the points.
- Draw the asymptote as a dashed line.
- Label the y-intercept.

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If your actual worksheet has different functions (e.g., shifted or transformed versions like \( y = 2^{x+1} \) or \( y = 2^x + 3 \)), please let me know, and I’ll adjust the solution accordingly.

Would you like me to generate a visual description or help you sketch these graphs?