Question

Suppose a parabola has vertex (6,5) and also passes through the point (7,7). Write the equation of the parabola in vertex form.

A. y=(x-6)^2+5
B. y=2(x-6)^2+5
C. y=2(x+6)^2+5
D. y=2(x-6)^2-5

Answer 1

Choice B:
`y = 2\, (x - 6)^(2) + 5`.

Explanation:

For a parabola with vertex
`(h,\, k)`, the vertex form equation of that parabola in would be:

`\text{$y = a\, (x - h)^(2) + k$ for some constant $a$}`.

In this question, the vertex is
`(6,\, 5)`, such that
`h = 6` and
`k = 5`. There would exist a constant
`a` such that the equation of this parabola would be:

`y = a\, (x - 6)^(2) + 5`.

The next step is to find the value of the constant
`a`.

Given that this parabola includes the point
`(7,\, 7)`,
`x = 7` and
`y = 7` would need to satisfy the equation of this parabola,
`y = a\, (x - 6)^(2) + 5`.

Substitute these two values into the equation for this parabola:

`7 = (7 - 6)^(2)\, a + 5`.

Solve this equation for
`a`:

`7 = a + 5`.

`a = 2`.

Hence, the equation of this parabola would be:

`y = 2\, (x - 6)^(2) + 5`.