How to find the equation for a parabola for which you are given two points and the vertex?

Question

I was originally given the value (4,-2) as the vertex of a parabola and told that it also includes the value (3,-5). From this point, I deduced that the next point would have the same y-value as the point whose x-value is equidistant from the vertex, so the next point would be (5,-5). I also know that since the parabola's vertex is higher than the two values surrounding it, it is a negative parabola. I am now stuck and do not know how to continue about finding the equation for this limited input/output table.

Answer 1

Your thinking is right on point!

 

f(x) = ax² + bx + c

f(4) = a(4²) +4x + c

-2 = 16a + 16b + c

 

f(3) = a(3²) + 3b + c

-5 = 9a + 3b + c

 

f(5) = a(5²) + 5b + c

-5 = 25a + 5b + c

 

Solve this system of equations for a, b, and c.

 

 

Answer 2

Use the vertex form of the parabola:

 

y = a(x-h)² + k

 

Where (h,k) is the location of the vertex.  Since the vertex is given as (4,-2):

 

y = a(x-4)² - 2

 

To find a, plug in the value of the known point (3,-5):

 

-5 = a(3-4)² - 2

-5 = a - 2

-3 = a

 

y = -3(x-4)² - 2

 

In standard form:

 

y = -3x2 + 24x -50