Write the equation in vertex form of the parabola that has the given axis of symmetry, points, and direction of opening.

Question

Axis of symmetry: x=3; points (1,9) (4,3); direction of opening: upward 

Answer 1

Vertex form: f (x) = a(x - h)² + k  where (h,k) is the coordinates of the vertex.

 

If we know the axis of symmetry, we know the x-value of the vertex. Therefore the vertex is now (3,k), and our equation is now:

f (x) = a(x - 3)² + k

 

 

We're also given two points. So we can use them in our equation to create a system:

 

3=a(4-3)²+k

9=a(1-3)²+k

 

simplifying each equation we get:

 

3=a+k

9=4a+k

 

If we subtract our equations we get:

 

-6=-3a

 

which gives us 2=a

 

 

Lastly we must solve for k. Substituting 2 in for a in our first equation of our system gives us:

3=2(4-3)²+k

 

Simplified we get:

3=2+k

 

which gives us 1=k

 

Now we just need to replace 2, 3 and 1 with a, h and k, respectively in our vertex form, giving us:

 

f (x) = 2(x - 3)² + 1