Question

Find the quadratic function that is the best fit for​ f(x) defined by the table below.
x
0
2
4
6
8
10

​f(x)
0
397
1602
3602
6402
9999
Question content area bottom
Part 1
The quadratic function is yequalsnothing x squared plus nothing x plus left parenthesis nothing right parenthesis
  
  
  
.
​(Type an integer or decimal rounded to two decimal places as​ needed.)

Answer 1

To find the quadratic function that best fits the given data, we will assume the function has the form:

\[ f(x) = ax^2 + bx + c \]

We have the data points:

\[
\begin{align*}
(0, 0) \\
(2, 397) \\
(4, 1602) \\
(6, 3602) \\
(8, 6402) \\
(10, 9999) \\
\end{align*}
\]

We'll create a system of equations based on these points to solve for \(a\), \(b\), and \(c\).

### Step 1: Set Up the System of Equations

Using the points, we can substitute into the equation \(f(x) = ax^2 + bx + c\):

1. For \(x = 0\), \(f(0) = 0\):
\[
c = 0
\]

2. For \(x = 2\), \(f(2) = 397\):
\[
4a + 2b + c = 397 \\
4a + 2b + 0 = 397 \quad (c = 0) \\
4a + 2b = 397 \quad \text{(Eq. 1)}
\]

3. For \(x = 4\), \(f(4) = 1602\):
\[
16a + 4b + c = 1602 \\
16a + 4b + 0 = 1602 \quad (c = 0) \\
16a + 4b = 1602 \quad \text{(Eq. 2)}
\]

4. For \(x = 6\), \(f(6) = 3602\):
\[
36a + 6b + c = 3602 \\
36a + 6b + 0 = 3602 \quad (c = 0) \\
36a + 6b = 3602 \quad \text{(Eq. 3)}
\]

### Step 2: Solve the System

First, we can eliminate \(b\) by solving Eq. 1 and Eq. 2:

From Eq. 1:
\[
2b = 397 - 4a \\
b = \frac{397 - 4a}{2}
\]

Substitute \(b\) into Eq. 2:
\[
16a + 4\left(\frac{397 - 4a}{2}\right) = 1602 \\
16a + 2(397 - 4a) = 1602 \\
16a + 794 - 8a = 1602 \\
8a = 1602 - 794 \\
8a = 808 \\
a = 101
\]

Now substitute \(a\) back into Eq. 1 to find \(b\):
\[
4(101) + 2b = 397 \\
404 + 2b = 397 \\
2b = 397 - 404 \\
2b = -7 \\
b = -3.5
\]

### Summary of the Coefficients
- \(a = 101\)
- \(b = -3.5\)
- \(c = 0\)

### Step 3: Write the Quadratic Function
Now we can write the quadratic function:
\[
f(x) = 101x^2 - 3.5x + 0
\]

So, the best-fit quadratic function is:

\[
f(x) = 101x^2 - 3.5x
\]

### Final Answer Format
The quadratic function in the requested format is:

\[ y = 101x^2 - 3.5x + 0 \]

The values to fill in are:
- \(a = 101\)
- \(b = -3.5\)
- \(c = 0\)

Write it as:

\[
y = 101x^2 - 3.5x + 0
\]