Polynomials

Question

find a polynomial function of degree 2 whose x intercept are -1 and 5 and whose range is (-∞,9)

 

Answer 1

A degree 2 polynomial is a quadratic equation whose graph is a parabola.  If the range is (-∞,9), then we have an inverted parabola with the vertex at the top.  The y-coordinate of the vertex is 9.  Use the vertex form of  the quadratic equation:

 

y = a(x-h)² + k

 

where a is a constant and (h,k) is the location of the vertex.  So k = 9 and so far we have:

 

y = a(x-h)² + 9

 

We need to find a and h.  h is the x-coordinate of the vertex.  It lies exactly half way between the x-intercepts, which are -1 and 5, a distance of 6 apart.  Half of 6 is 3, so the vertex's x-coordinate, h = -1 + 3 = 2.  Now our equation is:

 

y = a(x-2)² + 9

 

To find a, let's plug in one of the x-intercept locations, say (-1,0):

 

0 = a(-1-2)² + 9

0 = a(-3)² + 9

0 = 9a + 9

-9 = 9a

-1 = a

 

So the final equation in vertex form is:

 

y = -(x-2)² + 9

 

In standard form, it's:

 

y = -x² + 4x + 5

 

In factored form, it's

 

y = -(x-5)(x+1)