Consider the following points.
(2013, 4), (2014, 5), (2015, 9)
(a) Determine the polynomial function of least degree whose graph passes through the given points.
p(x) = 


To find the polynomial function of least degree that passes through the given points \((2013, 4), (2014, 5), (2015, 9)\), we need to determine a polynomial of degree \(2\) (a quadratic polynomial) because three points define a parabola (a quadratic function).

A general quadratic polynomial has the form:

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

We have three points, which give us a system of three equations:

1. For \(x = 2013\), \(y = 4\):
   \[
   a(2013)^2 + b(2013) + c = 4
   \]

2. For \(x = 2014\), \(y = 5\):
   \[
   a(2014)^2 + b(2014) + c = 5
   \]

3. For \(x = 2015\), \(y = 9\):
   \[
   a(2015)^2 + b(2015) + c = 9
   \]

Let's solve this system of equations for \(a\), \(b\), and \(c\).The solution to the system of equations gives us the values for \(a\), \(b\), and \(c\):

\[
a = \frac{3}{2}, \quad b = -\frac{12079}{2}, \quad c = 6079264
\]

Thus, the polynomial function of least degree that passes through the given points is:

\[
p(x) = \frac{3}{2}x^2 - \frac{12079}{2}x + 6079264
\]

This is the quadratic polynomial that fits all three points.

### Would you like more details, or do you have any questions?

Here are five related questions that expand on this topic:

1. How can we determine if a given polynomial fits additional points?
2. What is the geometric interpretation of the coefficients in a quadratic polynomial?
3. How do we find the vertex of a quadratic function?
4. What are other methods to determine a polynomial that passes through a set of points?
5. How would the solution change if an additional point was added?

**Tip:** Quadratic polynomials always graph as parabolas. Understanding their properties helps in solving various mathematical problems involving curves.

Consider the following points. (2013, 4), (2014, 5), (2015, 9) (a) Determine the polynomial function of least degree whose graph passes through the given points. p(x) = 

Learn how to determine the quadratic polynomial of least degree that passes through the points (2013, 4), (2014, 5), and (2015, 9). This problem involves quadratic equations, polynomial interpolation, and solving systems of equations. Suitable for students in Grades 10-12.

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