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
3604
6401
9996
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 points, we can denote the quadratic function in the standard form as:

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

Given the data points:

\[
\begin{align*}
(0, 0), \\
(2, 397), \\
(4, 1602), \\
(6, 3604), \\
(8, 6401), \\
(10, 9996)
\end{align*}
\]

We can set up a system of equations using the quadratic function for several \(x\) values. Let's use three points to create a system of equations. We can choose \( (0, 0) \), \( (2, 397) \), and \( (4, 1602) \):

1. For \( (0, 0) \):
\[
0 = a(0^2) + b(0) + c \Rightarrow c = 0
\]

2. For \( (2, 397) \):
\[
397 = a(2^2) + b(2) + c \Rightarrow 397 = 4a + 2b
\]

3. For \( (4, 1602) \):
\[
1602 = a(4^2) + b(4) + c \Rightarrow 1602 = 16a + 4b
\]

Now we have a system with the equations:

\[
\begin{align*}
1. \quad & 4a + 2b = 397 \quad \text{(1)} \\
2. \quad & 16a + 4b = 1602 \quad \text{(2)}
\end{align*}
\]

We can solve these equations step by step:

From the first equation, solve for \(b\):

\[
2b = 397 - 4a \quad \Rightarrow \quad b = \frac{397 - 4a}{2}
\]

Substituting this expression for \(b\) into the second equation:

\[
16a + 4\left(\frac{397 - 4a}{2}\right) = 1602
\]

This simplifies to:

\[
16a + 2(397 - 4a) = 1602
\]

Expanding gives:

\[
16a + 794 - 8a = 1602
\]

Combining like terms results in:

\[
8a + 794 = 1602
\]

Subtracting 794 from both sides:

\[
8a = 808 \quad \Rightarrow \quad a = 101
\]

Now substitute \(a = 101\) back into the equation for \(b\):

\[
b = \frac{397 - 4(101)}{2} = \frac{397 - 404}{2} = \frac{-7}{2} = -3.5
\]

Now we have:

- \(a = 101\)
- \(b = -3.5\)
- \(c = 0\)

Thus, the quadratic function that best fits the data is:

\[
f(x) = 101x^2 - 3.5x + 0
\]

In standard form, the quadratic function is:

\[
\boxed{101x^2 - 3.5x}
\]

If you need the function where \(c\) is explicitly included, it can be represented as:

\[
f(x) = 101x^2 - 3.50x + 0
\]