Find a function f whose graph is a parabola with the given vertex and that passes through the given point. vertex (−1, 7); point (−3, −5)

Question

What is a function that would work with the given vertex and point?

Answer 1

Originally from Lowellville... small world;

played in the Warren Junior Military band

and friend of mine went to Harding.

 

y = C(x - k)^2 + k  where (h,k) is the vertex

 

y = C(x - (-1))^2 + 7

 

y = C( x +1)^2 + 7

 

Now for (-3,-5)

 

-5 = C(-3 +1)^2 + 7

-5 = C(-2)^2 + 7

-5 = C(4) + 7

-12 = 4c

c = -3

 

The parabola is y = -3(x+1)^2+7

                          = -3(x^2+2x+1)+7

                          = -3x^2 - 6x + 4

 

As a check:   (-1,7)  -->  -3(-1)^2 - 6(-1) + 4 = -3(1) + 6 + 4 = -3 + 6 + 4 = 3 + 4 = 7 <--- yes, it works

                   (-3,-5) -->  -3(-3)^2 -6(-3) + 4 = -3(9) + 18 +4 = -27 + 18 + 4 = -9 + 5 = -5 <--- yes, it works

 

 By symmetry property of parabolas, the point (-3,5) in relation to the vertex (-1,7) is

    12 units up and 2 units to the right.

 

The reflection is then 2 more units to the right and 12 units down from there.....

  so (1,-5)

 

Plugging this in.... -3(1)^2 - 6(1) + 4 = -3(1) - 6(1) + 4 = -3 - 6+4 = -9 +4 = -5 <--- yes it works!!!

 

 

The parabola is y = -3(x+1)^2+7
= -3(x^2+2x+1)+7
= -3x^2 - 6x + 4