solve f(x) = anxn + an−1xn−1 + ...+ a2x2 + a1x + a0** using the points (560,200), (600,160), and (750,0)

To solve for the coefficients \( a_n, a_{n-1}, \dots, a_1, a_0 \) of the polynomial function \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \) using the given points \((560, 200)\), \((600, 160)\), and \((750, 0)\), we will consider a quadratic polynomial since we have three points. 

### Quadratic Polynomial Assumption:
Let's assume the polynomial is of the form:
\[
f(x) = ax^2 + bx + c
\]
where \(a\), \(b\), and \(c\) are the coefficients to be determined.

### Using the given points:
1. For the point \( (560, 200) \):
\[
200 = a(560)^2 + b(560) + c
\]

2. For the point \( (600, 160) \):
\[
160 = a(600)^2 + b(600) + c
\]

3. For the point \( (750, 0) \):
\[
0 = a(750)^2 + b(750) + c
\]

### System of Equations:
This gives us the following system of equations:
\[
\begin{aligned}
200 &= a(560)^2 + b(560) + c \\
160 &= a(600)^2 + b(600) + c \\
0 &= a(750)^2 + b(750) + c
\end{aligned}
\]

We can solve this system of equations to find the values of \( a \), \( b \), and \( c \). I'll calculate these values now.The coefficients for the quadratic polynomial \( f(x) = ax^2 + bx + c \) based on the given points are:

\[
a = -\frac{1}{2850}, \quad b = -\frac{169}{285}, \quad c = \frac{12200}{19}
\]

So, the polynomial function is:

\[
f(x) = -\frac{1}{2850}x^2 - \frac{169}{285}x + \frac{12200}{19}
\]

Would you like further details on the solution process or have any other questions?

Here are some related questions you might find useful:

1. How can you solve a polynomial with more than three points?
2. What does the discriminant of a quadratic equation represent?
3. Can you use the quadratic formula to solve this equation?
4. How does the sign of coefficient \( a \) affect the shape of the parabola?
5. What is the vertex form of a quadratic equation?
6. How can you find the roots of a quadratic equation using this function?
7. What is the significance of the constant term in the quadratic equation?
8. How do you determine the axis of symmetry for this quadratic function?

**Tip:** When solving a system of equations, it’s helpful to write each equation clearly to avoid confusion and to systematically eliminate variables.

Learn how to solve a polynomial function using given points (560, 200), (600, 160), and (750, 0). This problem covers polynomial functions, quadratic equations, and systems of equations. Suitable for Grades 10-12.

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