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The graph shows an exponential curve with key points labeled: (0, 0.25), (2, 1), (3, 2), (4, 4), and (5, 8). This indicates that the relationship between \(x\) and \(y\) follows an exponential pattern.

We can use the general form of an exponential equation:
\[
y = a \cdot b^x
\]
where \(a\) is the initial value when \(x = 0\), and \(b\) is the base of the exponent. From the point (0, 0.25), we can deduce that \(a = 0.25\), since when \(x = 0\), \(y = 0.25\).

Looking at the ratio between points:
- From \(x = 0\) to \(x = 1\), the value of \(y\) doubles.
- From \(x = 2\) to \(x = 3\), the same doubling occurs.

This suggests that the base \(b = 2\), as the function doubles with each step of \(x\).

Thus, the equation could be written as:
\[
y = 0.25 \cdot 2^x
\]

To match the options provided in the multiple-choice question:
- \( y = 0.25 \cdot 2^x \) matches the option \( y = 4 \cdot 2^{x - 4} \), which is the same function rewritten with a shift.

Now, let's match other forms:
1. **\( y = 1.5(2^x) \)** — Incorrect, since the initial value is not 1.5.
2. **\( y = 2(2^{x-1}) \)** — Incorrect, as it shifts and starts from \( y = 2 \).
3. **\( y = 8(2^{x-5}) \)** — This is a correct transformation, as it's just a shifted version of the curve, starting with \( y = 8 \) at \( x = 5 \).
4. **\( y = 2(2^{x-3}) \)** — This is also valid, corresponding to a shift starting at \( x = 3 \) with \( y = 2 \).
5. **\( y = 0(2^{x-0.25}) \)** — Incorrect, as this would make the equation equal to 0 for all values of \(x\).
6. **\( y = 4(2^{x-4}) \)** — This is also a valid choice, as it's another shift of the same exponential function.

### Correct answers: 
- \( y = 8(2^{x-5}) \)
- \( y = 2(2^{x-3}) \)
- \( y = 4(2^{x-4}) \)

Would you like further details or clarifications?

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Here are 5 related questions for practice:
1. How do you determine the base of an exponential function from a graph?
2. What is the significance of the initial value in an exponential function?
3. How can you identify shifts in exponential graphs?
4. What happens to an exponential graph when you increase or decrease the base?
5. How do transformations like \( y = a \cdot b^{x - c} \) affect the graph?

**Tip:** Exponential functions grow (or decay) rapidly because the rate of change increases as \(x\) increases. Recognizing the doubling (or halving) pattern helps identify the base of the exponent.

Which equations could represent the graph in point-ratio form? The graph shows an exponential curve with key points (0, 0.25), (2, 1), (3, 2), (4, 4), and (5, 8).

Explore how to determine the correct exponential equation from a graph with points (0, 0.25), (2, 1), (3, 2), (4, 4), and (5, 8). This problem focuses on identifying exponential patterns and matching them to point-ratio forms. Suitable for students in Grades 9-12.

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