How do i find the equation of a parabola given 2 points and the axis of symmetry, but no vertex?

Question

I have that it passes through 1,4 and 2,7 and that it has axis of symmetry at x = 0, but nothing else

Answer 1

Using the vertex form of a parabola f(x) = a(x - h)² + k where (h,k) is the vertex of the parabola

The axis of symmetry is x = 0 so h also equals 0

Substitute each point from the parabola into the vertex form:

4 = a(1 - 0)² + k

4 = a(1) + k

4 = a + k

7 = a(2 - 0)² + k

7 = a(4) + k

7 = 4a + k

We know have a linear system:

4 = a + k

7 = 4a + k

Subtracting the two equations gives us:

-3 = -3a

a = 1

Substituting the a value into the first equation of the linear system:

4 = 1 + k

k = 3

f(x) = (x - 0)² + 3

f(1) = 4 = (1 - 0)² + 3 = 1 + 3

f(2) = 7 = (2 - 0)² + 3 = 4 + 3

The equation of the parabola through the given points and axis of symmetry is

f(x) = (x - 0)² + 3 = x² + 3

Answer 2

4 = a(x - 1) + k

7 = a(x - 2) + k

Solve for a, i.e., eliminate k

Substitute either point and into either equation with "a" value and solve for k.