Writing Exponential Equations Using a Graph - Problems involving exponential functions and rate of change.

Since I can't view or access uploaded images directly, I can’t see the specific graphs you're referring to. However, I can guide you through how to solve problems like this based on the structure of your worksheet.

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Objective:

For each graph, determine the exponential function in the form:

$$
f(x) = ab^x
$$

Then state:
- The domain
- The range
- The rate of change over a given interval (usually [0, 1], [0, 2], etc.)

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## 🔍 Step-by-Step Guide

Let’s go through how to analyze one of these graphs and find the equation.

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✔ Step 1: Identify Key Points on the Graph

Look at the graph and find two points that lie exactly on the curve. Usually, the y-intercept is easy to spot — it's where $ x = 0 $. That gives you $ f(0) = a $, because:

$$
f(0) = ab^0 = a \cdot 1 = a
$$

So:
- Find $ f(0) $ → this is $ a $
- Find another point, say $ f(1) $, $ f(2) $, etc.
- Use that to solve for $ b $

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✔ Step 2: Solve for $ b $

Suppose:
- $ f(0) = a $
- $ f(1) = ab $

Then:
$$
b = \frac{f(1)}{a}
$$

Or if you have $ f(2) = ab^2 $, then:
$$
b^2 = \frac{f(2)}{a} \Rightarrow b = \sqrt{\frac{f(2)}{a}}
$$

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✔ Step 3: Write the Equation

Once you know $ a $ and $ b $, write:
$$
f(x) = ab^x
$$

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✔ Step 4: Domain and Range

For exponential functions $ f(x) = ab^x $:

- Domain: All real numbers → $ (-\infty, \infty) $
- Range:
- If $ a > 0 $ and $ b > 1 $: $ (0, \infty) $
- If $ a > 0 $ and $ 0 < b < 1 $: $ (0, \infty) $
- If $ a < 0 $: range is $ (-\infty, 0) $
- But since most graphs start above x-axis, assume $ a > 0 $, so range is $ (0, \infty) $

> Note: If the graph has a horizontal asymptote (like y = 0), that supports the range being $ (0, \infty) $

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✔ Step 5: Rate of Change

The rate of change over an interval $[x_1, x_2]$ is:

$$
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
$$

But in this context, they likely want the average rate of change over intervals like [0,1], [0,2], etc.

Also, which interval produces the greatest rate of change?

👉 Exponential functions grow faster as $ x $ increases (if $ b > 1 $), so the largest interval with higher x-values will generally have the largest average rate of change.

But let’s test it with values.

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## 📌 Example Problem (Based on Common Graphs)

Let’s suppose Graph 1 looks like this:

- Passes through $ (0, 1) $
- Passes through $ (1, 2) $

Then:
- $ f(0) = 1 \Rightarrow a = 1 $
- $ f(1) = 2 \Rightarrow ab = 2 \Rightarrow b = 2 $

So:
$$
f(x) = 2^x
$$

- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $

Now compute rate of change:

- Over $[0,1]$: $ \frac{f(1) - f(0)}{1 - 0} = \frac{2 - 1}{1} = 1 $
- Over $[0,2]$: $ f(2) = 4 $, so $ \frac{4 - 1}{2} = 1.5 $
- Over $[0,3]$: $ f(3) = 8 $, so $ \frac{8 - 1}{3} \approx 2.33 $

So [0,3] has the greatest rate of change.

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## ⚠️ Important Notes:

- If the graph is decreasing, then $ 0 < b < 1 $, e.g., $ f(x) = (1/2)^x $
- If the graph starts at $ (0, 3) $ and goes through $ (1, 6) $, then $ a = 3 $, $ b = 2 $, so $ f(x) = 3 \cdot 2^x $
- Always check the y-intercept first!

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## 🧩 Now Apply This to Your Graphs

Even without seeing the image, here’s what you should do for each graph:

For each graph (1–6):

#### Step 1: Find two points
- Look at $ x = 0 $: What is $ y $? That’s $ a $
- Find $ x = 1 $: What is $ y $? Then $ b = \frac{y}{a} $

#### Step 2: Write equation
$ f(x) = ab^x $

#### Step 3: Domain & Range
- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $ (unless shifted — but these look basic)

#### Step 4: Compute Rate of Change
Use:
- $ \frac{f(1) - f(0)}{1 - 0} $
- $ \frac{f(2) - f(0)}{2 - 0} $
- $ \frac{f(3) - f(0)}{3 - 0} $

Compare which interval has the largest value.

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## ❓ How to Handle the "Rate of Change" Table?

You’re asked:

> Which interval produces the greatest rate of change?

So for each graph, calculate the average rate of change over:
- [0,1]
- [0,2]
- [0,3]

Then pick the one with the highest value.

Because exponential growth accelerates, the [0,3] interval usually has the highest average rate of change (even though it's averaged over more time).

But sometimes if the function grows slowly, [0,1] might be bigger — but typically not.

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## ✔ Final Answer Format (Example)

Let’s say Graph 1 has:
- Points: (0, 1), (1, 3)
- So $ a = 1 $, $ b = 3 $
- $ f(x) = 3^x $
- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $
- Rate of change:
- [0,1]: $ \frac{3 - 1}{1} = 2 $
- [0,2]: $ \frac{9 - 1}{2} = 4 $
- [0,3]: $ \frac{27 - 1}{3} \approx 8.67 $
- Greatest rate of change: [0,3]

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📝 Summary

| Graph | Equation | Domain | Range | Greatest Rate of Change |
|-------|---------|--------|-------|--------------------------|
| 1 | $ f(x) = 2^x $ | $ (-\infty, \infty) $ | $ (0, \infty) $ | [0,3] |
| 2 | $ f(x) = 3^x $ | $ (-\infty, \infty) $ | $ (0, \infty) $ | [0,3] |
| ... | ... | ... | ... | ... |

> You’ll need to adjust based on actual points from your graphs.

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🛠️ Tips:

- If the graph passes through $ (0, 2) $ and $ (1, 6) $, then $ a = 2 $, $ b = 3 $ → $ f(x) = 2 \cdot 3^x $
- If it passes through $ (0, 4) $ and $ (1, 2) $, then $ a = 4 $, $ b = 0.5 $ → $ f(x) = 4 \cdot (0.5)^x $
- Check whether the graph is growing or decaying.

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If you can describe the graphs (e.g., “Graph 1 goes through (0,1) and (1,2)” or “Graph 4 starts at (0,3) and doubles every step”), I can help you solve them specifically.

Just tell me the points from each graph!