To find the corresponding \( y \)-value when \( x = 2.0 \), we can use interpolation since we have three given points:

1. \( (0, 0) \)
2. \( (1.5, 2.3) \)
3. \( (2.5, 6.3) \)

We will use linear interpolation between the two points closest to \( x = 2.0 \), which are \( (1.5, 2.3) \) and \( (2.5, 6.3) \).

### Linear Interpolation Formula

For two points \((x_1, y_1)\) and \((x_2, y_2)\), the interpolation formula to find \( y \) at \( x \) is:

\[
y = y_1 + \frac{(y_2 - y_1)}{(x_2 - x_1)} \cdot (x - x_1)
\]

### Applying the Formula

1. \( (x_1, y_1) = (1.5, 2.3) \)
2. \( (x_2, y_2) = (2.5, 6.3) \)
3. \( x = 2.0 \)

Substitute these values into the interpolation formula:

\[
y = 2.3 + \frac{(6.3 - 2.3)}{(2.5 - 1.5)} \cdot (2.0 - 1.5)
\]

\[
y = 2.3 + \frac{4.0}{1.0} \cdot 0.5
\]

\[
y = 2.3 + 4.0 \cdot 0.5
\]

\[
y = 2.3 + 2.0
\]

\[
y = 4.3
\]

So, when \( x = 2.0 \), the corresponding \( y \)-value is **4.3**.

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Would you like more details or have any questions?

### Related Questions:

1. How would you perform quadratic interpolation using all three points to find \( y \) at \( x = 2.0 \)?
2. How does interpolation differ from extrapolation, and what are the risks of using one over the other?
3. If you were to use a polynomial function to fit all three points, what would be the degree of the polynomial, and how would you determine it?
4. How can you use the method of least squares to find the best-fitting line or curve through these points?
5. How would you implement this interpolation calculation using a programming language like Python?

### Tip:

When using interpolation, ensure that the points you are using are close to the value you're interpolating for, as this increases accuracy and reliability.

Learn how to use linear interpolation to find the y-value when x=2.0, given the points (0,0), (1.5, 2.3), and (2.5, 6.3). This detailed solution walks through the interpolation formula, step-by-step calculations, and is ideal for students in Grades 9-12.

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