Help, my math book doesn't tell me how to solve this?

Does your text tell you what the vertex form of a parabola is?

y = a(x-g)² + h : parabola with vertical axis

x = a(y-h)² + g : parabola with horizontal axis

You are given the vertex: (g, h) = (1,3). Substitute those values

Vertical axis:

y = a(x-1)² + 3

Horizontal axis:

x = a(y - 3)² + 1

You are given a point on the locus: (x, y) = (2,5)

Vertical Axis:

5 = a(2-1)² + 3

2 = a

Horizontal Axis:

2 = a(5-3)² + 1

1 = 4a

a = 1/4

The parabola is:

Vertical Axis: y = 2(x-1)² + 3

Horizontal Axis: x = (y-3)²/4 + 1

There are also an infinity of other parabolas with a vertex at (1,3) passing through the point (2,5), each with a different rotation angle.

The equation of a parabola is

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y = a(x - h)² + k

Let's substitute the vertex on the equation

-----------------------------

y = a(x - 1)² + 3

Let's substitute the point on the equation to find "a"

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5 = a(2 - 1)² + 3

5 - 3 = a

a = 2

Therefore, the equation is

-----------------------------

y = 2(x - 1)² + 3

plug the vertex points it into the vertex form equation

y=a(x-1)^2+3

to solve for a plug in (2,5)

5=a(2-1)^2+3

5=a(1)^2+3

5=1a+3

-3 -3

2=1a

2=a

so your final answer would be

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

Vertex form:

y = a(x - h)² + k, where (h,k) is the vertex. Vertex is at (1,3):

y = a(x - 1)² + 3; Now we can sub in point (2,5) for x and y and solve for a:

5 = a(2 - 1)² + 3

5 = a(1)² + 3

5 = a + 3

a = 2

so:

y = 2(x - 1)² + 3

y = a * (x - h)^2 + k

(h , k) is the vertex

(x , y) is a point on the parabola

y = a * (x - 1)^2 + 3

5 = a * (2 - 1)^2 + 3

2 = a * 1^2

2 = a

y = 2 * (x - 1)^2 + 3

That's it.