Exponential pattern in input-output table with multiplication by 2.

Let's carefully analyze the problem shown in the image and solve it step by step.

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Given:

We have a table of input-output values:

| Input (x) | Output (y) |
|-----------|------------|
| -1 | 1.5 |
| 0 | 3 |
| 1 | 6 |
| 2 | 12 |
| 3 | 24 |
| 4 | 48 |
| 5 | 96 |

We are to determine the pattern and find a formula that relates the input $ x $ to the output $ y $.

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Step 1: Check for a Linear Pattern

The image shows first differences and second differences:

- First differences (change in output):
- $ 3 - 1.5 = 1.5 $
- $ 6 - 3 = 3 $
- $ 12 - 6 = 6 $
- $ 24 - 12 = 12 $
- $ 48 - 24 = 24 $
- $ 96 - 48 = 48 $

So, first differences:
1.5, 3, 6, 12, 24, 48

Now second differences:
- $ 3 - 1.5 = 1.5 $
- $ 6 - 3 = 3 $
- $ 12 - 6 = 6 $
- $ 24 - 12 = 12 $
- $ 48 - 24 = 24 $

Second differences:
1.5, 3, 6, 12, 24 — these are not constant, so not linear.

But notice: the second differences follow the same pattern as the first differences, suggesting an exponential pattern.

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Step 2: Look for Multiplicative Pattern

Instead of adding, let’s check if each output is multiplied by a constant factor.

Check ratios between successive outputs:

- $ \frac{3}{1.5} = 2 $
- $ \frac{6}{3} = 2 $
- $ \frac{12}{6} = 2 $
- $ \frac{24}{12} = 2 $
- $ \frac{48}{24} = 2 $
- $ \frac{96}{48} = 2 $

✔ Every time the input increases by 1, the output doubles → common ratio = 2

This confirms an exponential function of the form:

$$
y = ab^x
$$

Where:
- $ b = 2 $ (since output doubles with each increase in x)
- We need to find $ a $ using one known point.

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Step 3: Find the Base Function

Use the point $ x = 0 $, $ y = 3 $:

$$
y = a \cdot 2^x
$$
$$
3 = a \cdot 2^0 = a \cdot 1 \Rightarrow a = 3
$$

So the equation is:

$$
y = 3 \cdot 2^x
$$

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Step 4: Verify with Other Points

Let’s test this formula:

- $ x = -1 $: $ y = 3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = 1.5 $ ✔
- $ x = 0 $: $ y = 3 \cdot 2^0 = 3 \cdot 1 = 3 $ ✔
- $ x = 1 $: $ y = 3 \cdot 2^1 = 6 $ ✔
- $ x = 2 $: $ y = 3 \cdot 2^2 = 3 \cdot 4 = 12 $ ✔
- $ x = 3 $: $ y = 3 \cdot 2^3 = 3 \cdot 8 = 24 $ ✔
- $ x = 4 $: $ y = 3 \cdot 2^4 = 3 \cdot 16 = 48 $ ✔
- $ x = 5 $: $ y = 3 \cdot 2^5 = 3 \cdot 32 = 96 $ ✔

All match!

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Final Answer:

The rule is:

$$
\boxed{y = 3 \cdot 2^x}
$$

This is an exponential function with base 2 and initial value 3 at $ x = 0 $. The output doubles every time the input increases by 1.

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Why the Image Suggests This:

- The first differences grow exponentially, which is a hallmark of exponential functions.
- The ratios are constant, confirming multiplicative growth.
- The note says “this time the pattern repeats” — referring to the doubling behavior — which is typical of exponential functions.
- The multiplier (2) is the base of the exponential function.

✔ So, the solution is confirmed.