https://file.aiquickdraw.com/mathbot/file/17372803344083ck47b07.jpg

Use langrange method and the answer is y= (3/8)x^2- (11/4)x+6 and y(5)= 13/8

Thank you for uploading the question. Let's solve the problem using the **Lagrange interpolation method** to confirm the polynomial and find \( y(5) \).

### Given Data:
| \( x \) | \( y \) |
|--------|--------|
| 4      | 1      |
| 6      | 3      |
| 8      | 8      |
| 10     | 16     |

The formula for the Lagrange polynomial is:

\[
P(x) = \sum_{i=1}^{n} y_i \cdot \ell_i(x)
\]

where:
\[
\ell_i(x) = \prod_{j=1, j \neq i}^{n} \frac{x - x_j}{x_i - x_j}
\]

### Step 1: Compute the polynomial
We will compute the Lagrange basis polynomials \( \ell_1(x), \ell_2(x), \ell_3(x), \ell_4(x) \) for \( x_1 = 4, x_2 = 6, x_3 = 8, x_4 = 10 \), and substitute the corresponding \( y_i \) values.

---

#### 1. Lagrange basis \( \ell_1(x) \):
\[
\ell_1(x) = \frac{(x - 6)(x - 8)(x - 10)}{(4 - 6)(4 - 8)(4 - 10)} = \frac{(x - 6)(x - 8)(x - 10)}{(-2)(-4)(-6)} = -\frac{1}{48}(x - 6)(x - 8)(x - 10)
\]

#### 2. Lagrange basis \( \ell_2(x) \):
\[
\ell_2(x) = \frac{(x - 4)(x - 8)(x - 10)}{(6 - 4)(6 - 8)(6 - 10)} = \frac{(x - 4)(x - 8)(x - 10)}{(2)(-2)(-4)} = \frac{1}{16}(x - 4)(x - 8)(x - 10)
\]

#### 3. Lagrange basis \( \ell_3(x) \):
\[
\ell_3(x) = \frac{(x - 4)(x - 6)(x - 10)}{(8 - 4)(8 - 6)(8 - 10)} = \frac{(x - 4)(x - 6)(x - 10)}{(4)(2)(-2)} = -\frac{1}{16}(x - 4)(x - 6)(x - 10)
\]

#### 4. Lagrange basis \( \ell_4(x) \):
\[
\ell_4(x) = \frac{(x - 4)(x - 6)(x - 8)}{(10 - 4)(10 - 6)(10 - 8)} = \frac{(x - 4)(x - 6)(x - 8)}{(6)(4)(2)} = \frac{1}{48}(x - 4)(x - 6)(x - 8)
\]

---

### Step 2: Combine the terms
The polynomial is:
\[
P(x) = y_1 \cdot \ell_1(x) + y_2 \cdot \ell_2(x) + y_3 \cdot \ell_3(x) + y_4 \cdot \ell_4(x)
\]

Substitute \( y_1 = 1, y_2 = 3, y_3 = 8, y_4 = 16 \):

\[
P(x) = 1 \cdot \left[ -\frac{1}{48}(x - 6)(x - 8)(x - 10) \right]
+ 3 \cdot \left[ \frac{1}{16}(x - 4)(x - 8)(x - 10) \right]
\]
\[
+ 8 \cdot \left[ -\frac{1}{16}(x - 4)(x - 6)(x - 10) \right]
+ 16 \cdot \left[ \frac{1}{48}(x - 4)(x - 6)(x - 8) \right]
\]

Expanding and simplifying will give:
\[
P(x) = \frac{3}{8}x^2 - \frac{11}{4}x + 6
\]

---

### Step 3: Find \( y(5) \):
Substitute \( x = 5 \) into \( P(x) \):
\[
P(5) = \frac{3}{8}(5^2) - \frac{11}{4}(5) + 6
\]
\[
P(5) = \frac{3}{8}(25) - \frac{55}{4} + 6 = \frac{75}{8} - \frac{110}{8} + \frac{48}{8}
\]
\[
P(5) = \frac{13}{8}
\]

---

### Final Answer:
The polynomial is:
\[
P(x) = \frac{3}{8}x^2 - \frac{11}{4}x + 6
\]

The value of \( y(5) \) is:
\[
y(5) = \frac{13}{8}
\]

---

Would you like further explanation or details?  

---

### Related Questions:
1. How is Lagrange interpolation used in other fields of study?
2. What are the limitations of the Lagrange interpolation method?
3. How can you verify that the polynomial passes through all given points?
4. How do higher-degree polynomials affect the accuracy of interpolation?
5. Can this method be used for data with uneven intervals in \( x \)?

---

### Tip:
For large datasets, Lagrange interpolation becomes computationally intensive; consider using Newton's divided difference method for efficiency.

Construct a polynomial for the data given below using the Lagrange interpolation method and find y when x = 5. Data: x = [4, 6, 8, 10] and y = [1, 3, 8, 16].

Discover how to construct a polynomial using the Lagrange interpolation method for given data points: x = [4, 6, 8, 10] and y = [1, 3, 8, 16]. Learn how to calculate the polynomial P(x) = (3/8)x^2 - (11/4)x + 6 and find y(5) = 13/8 with detailed step-by-step explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation