Exponential Functions Graphing Practice B worksheet with graphs and tables for function analysis.

It looks like you've uploaded a worksheet titled "Exponential Functions Graphing Practice B", which includes several exponential functions and their corresponding graphs. The task is to create a table of values for each function and then match it to the correct graph.

Let’s go through this step by step, analyzing each problem and explaining how to solve it.

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🔹 Problem 7: $ f(x) = 2^{-x} $

#### Step 1: Understand the Function
This is an exponential decay function because the base (2) is greater than 1, but the exponent is negative: $ -x $. So:
$$
f(x) = 2^{-x} = \frac{1}{2^x}
$$
As $ x $ increases, $ f(x) $ decreases toward 0; as $ x $ decreases (goes negative), $ f(x) $ grows rapidly.

#### Step 2: Create a Table of Values

| $ x $ | $ f(x) = 2^{-x} $ |
|--------|---------------------|
| -3 | $ 2^{3} = 8 $ |
| -2 | $ 2^{2} = 4 $ |
| -1 | $ 2^{1} = 2 $ |
| 0 | $ 2^{0} = 1 $ |
| 1 | $ 2^{-1} = 0.5 $ |
| 2 | $ 2^{-2} = 0.25 $ |
| 3 | $ 2^{-3} = 0.125 $|

#### Step 3: Match to Graph
Look at the graph on the left side (top-left). It shows:
- A curve starting high on the left (large y-values when x is negative)
- Decreasing as x increases
- Approaching 0 as x → ∞
- Passing through (0,1)

✔ This matches Graph 7 — correctly matched.

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🔹 Problem 8: $ f(x) = 2^{0.5x} $

This is an exponential growth function with a slower rate due to the coefficient $ 0.5 $ in the exponent.

$$
f(x) = 2^{0.5x} = (\sqrt{2})^x \approx (1.414)^x
$$

#### Step 2: Table of Values

| $ x $ | $ f(x) = 2^{0.5x} $ |
|--------|----------------------|
| -4 | $ 2^{-2} = 0.25 $ |
| -2 | $ 2^{-1} = 0.5 $ |
| 0 | $ 2^{0} = 1 $ |
| 2 | $ 2^{1} = 2 $ |
| 4 | $ 2^{2} = 4 $ |
| 6 | $ 2^{3} = 8 $ |

Note: At $ x=6 $, $ f(6) = 2^{3} = 8 $

#### Step 3: Match to Graph
The graph shown (top-right) starts low, increases slowly, passes through (0,1), and rises gradually. It's not doubling every unit, but rather growing more slowly.

✔ Matches Graph 8 — correctly matched.

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🔹 Problem 9: $ f(x) = 3^{-x} $

This is similar to Problem 7, but with base 3 instead of 2.

$$
f(x) = 3^{-x} = \frac{1}{3^x}
$$

#### Step 2: Table of Values

| $ x $ | $ f(x) = 3^{-x} $ |
|--------|---------------------|
| -3 | $ 3^{3} = 27 $ |
| -2 | $ 3^{2} = 9 $ |
| -1 | $ 3^{1} = 3 $ |
| 0 | $ 3^{0} = 1 $ |
| 1 | $ 3^{-1} = 1/3 \approx 0.333 $ |
| 2 | $ 3^{-2} = 1/9 \approx 0.111 $ |
| 3 | $ 3^{-3} = 1/27 \approx 0.037 $ |

#### Step 3: Match to Graph
The graph (bottom-left) shows:
- High values when $ x $ is negative
- Decreasing rapidly as $ x $ increases
- Passes through (0,1)
- Asymptotic to x-axis

But compare to Problem 7: Since base 3 > base 2, this decays faster than $ 2^{-x} $.

So $ 3^{-x} $ drops faster than $ 2^{-x} $.

✔ This matches Graph 9 — correctly matched.

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🔹 Problem 10: $ f(x) = 10^{1.5x} $

This is a fast-growing exponential function with base 10 and exponent scaled by 1.5.

$$
f(x) = 10^{1.5x} = (10^{1.5})^x \approx (31.62)^x
$$

#### Step 2: Table of Values

| $ x $ | $ f(x) = 10^{1.5x} $ |
|--------|------------------------|
| -2 | $ 10^{-3} = 0.001 $ |
| -1 | $ 10^{-1.5} = 10^{-1} \cdot 10^{-0.5} = 0.1 \cdot \frac{1}{\sqrt{10}} \approx 0.1 \cdot 0.316 = 0.0316 $ |
| 0 | $ 10^{0} = 1 $ |
| 1 | $ 10^{1.5} = 10 \cdot \sqrt{10} \approx 10 \cdot 3.162 = 31.62 $ |
| 2 | $ 10^{3} = 1000 $ |

Wait — let's compute carefully:

- $ f(-2) = 10^{1.5 \cdot (-2)} = 10^{-3} = 0.001 $
- $ f(-1) = 10^{-1.5} = 10^{-1} \cdot 10^{-0.5} = 0.1 \cdot \frac{1}{\sqrt{10}} \approx 0.1 \cdot 0.3162 = 0.03162 $
- $ f(0) = 1 $
- $ f(1) = 10^{1.5} = \sqrt{10^3} = \sqrt{1000} \approx 31.62 $
- $ f(2) = 10^{3} = 1000 $

So:

| $ x $ | $ f(x) $ |
|--------|-------------------|
| -2 | 0.001 |
| -1 | ~0.0316 |
| 0 | 1 |
| 1 | ~31.62 |
| 2 | 1000 |

#### Step 3: Match to Graph
The bottom-right graph shows:
- Starts near zero when $ x $ is negative
- Increases very rapidly
- Passes through (0,1)
- Grows extremely fast after $ x=1 $

This matches the behavior of $ f(x) = 10^{1.5x} $

✔ Matches Graph 10 — correctly matched.

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✔ Summary: All Answers Are Correct!

Each function has been matched to its correct graph based on:
- Behavior: Growth or decay?
- Rate of change: How fast does it grow/decay?
- Key points: Does it pass through (0,1)? What happens as $ x \to \infty $ or $ x \to -\infty $?

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Final Answer:

All four problems are already correctly matched in the image. The tables and graphs align properly:

- Problem 7: $ f(x) = 2^{-x} $ → Graph showing decay from left to right, passing through (0,1), asymptote at y=0.
- Problem 8: $ f(x) = 2^{0.5x} $ → Slow growth, passes through (0,1), increases gradually.
- Problem 9: $ f(x) = 3^{-x} $ → Faster decay than $ 2^{-x} $, same shape but steeper decline.
- Problem 10: $ f(x) = 10^{1.5x} $ → Rapid growth, passes through (0,1), explodes quickly.

✔ All answers are correct as presented.

Let me know if you'd like help creating your own table or graphing these manually!