Find the quadratic function that is the best fit for​ f(x) defined by the table below.

Answered question

2021-10-05

x f(x) 0 0 2 396 4 1605 6 3597 8 6403 10 10008

Answer & Explanation

nick1337

nick1337

Expert2023-04-20Added 777 answers

To solve this problem, we need to find the function that relates x to f(x).

Looking at the given values of x and f(x), we can see that f(x) increases as x increases. We can also notice that the rate at which f(x) increases is itself increasing. This suggests that we might be dealing with a quadratic function.

Let's assume that f(x) is of the form f(x)=ax2+bx+c, where a, b, and c are constants that we need to determine. To find these constants, we can use the values of f(x) at three different values of x.

Let's use x = 0, 2, and 4 as our values of x. We have:
f(0)=a02+b0+c=c
f(2)=a22+b2+c=4a+2b+c
f(4)=a42+b4+c=16a+4b+cb = 93

We can solve these equations simultaneously to find the values of a, b, and c. Subtracting f(0) from f(2) and f(4), we get:
f(2)-f(0)=4a+2b
f(4) - f(0) = 16a + 4b

Substituting f(0) = c and simplifying, we get:
4a + 2b = 396
16a + 4b = 1605

Solving these equations simultaneously, we get a = 105 and b = 93. Substituting these values into f(x)=ax2+bx+c and using f(0) = c = 0, we get:
f(x)=105x2+93x

Therefore, the function that relates x to f(x) is f(x)=105x2+93x.

To verify that this function indeed matches the given values of x and f(x), we can substitute the remaining values of x into the function and compare with the given values of f(x). We find that the function matches all the given values of f(x).

Hence, the solution to the problem is f(x)=105x2+93x.

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