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
3601
6401
9998
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 best fit quadratic function for the data given, we can use the general form of a quadratic function:

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

We'll need to determine the coefficients \( a \), \( b \), and \( c \) using methods such as polynomial regression. However, I can guide you through the process of setting up the equations based on the data provided to find these coefficients.

Given your data at the points:

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 0 \\
2 & 397 \\
4 & 1602 \\
6 & 3601 \\
8 & 6401 \\
10 & 9998 \\
\hline
\end{array}
\]

We can set up a system of equations by plugging the \( x \) values into the quadratic function form. From the first three data points, we can derive the following equations:

1. When \( x = 0 \):
\[ c = 0 \]

2. When \( x = 2 \):
\[ 4a + 2b + c = 397 \]

3. When \( x = 4 \):
\[ 16a + 4b + c = 1602 \]

Using \( c = 0 \), we can simplify these to:

1. \( 4a + 2b = 397 \) (Equation 1)
2. \( 16a + 4b = 1602 \) (Equation 2)

Now, we can solve Equation 1 for \( b \):

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

Substituting this expression for \( b \) into Equation 2:

\[
16a + 4\left(\frac{397 - 4a}{2}\right) = 1602
\]
\[
16a + 2(397 - 4a) = 1602
\]
\[
16a + 794 - 8a = 1602
\]
\[
8a + 794 = 1602
\]
\[
8a = 808
\]
\[
a = 101
\]

Now substituting \( a \) back into Equation 1 to find \( b \):

\[
4(101) + 2b = 397
\]
\[
404 + 2b = 397
\]
\[
2b = 397 - 404
\]
\[
2b = -7
\]
\[
b = -3.5
\]

Now we have \( a \), \( b \), and \( c \):

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

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

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

In the format requested, this can be expressed as:

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

So, the final answer is:
- \( a = 101 \)
- \( b = -3.50 \)
- \( c = 0 \)

So the quadratic function is:

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