Question

What is the equation of the parabola, in vertex form, that contains the point (2, -1) and has (4, -3) as a vertex?

a) (y = (x - 4)^2 - 3)
b) (y = (x - 2)^2 - 1)
c) (y = (x + 4)^2 - 3)
d) (y = (x + 2)^2 - 1)

Answer 1

The equation of the parabola with vertex (4, -3) that contains the point (2, -1) is y = 0.5(x - 4)^2 - 3, which corresponds to option a) (y = (x - 4)^2 - 3).

Step-by-step explanation:

The equation of a parabola in vertex form is given as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, the vertex of the parabola is given as (4, -3). Therefore, the equation will have the form y = a(x - 4)^2 - 3. To find the value of 'a', we can use the point (2, -1) that lies on the parabola. Substituting these coordinates into the equation gives us:

-1 = a(2 - 4)^2 - 3

Adding 3 to both sides gives us 2 = a(2 - 4)^2, which simplifies to 2 = 4a. Dividing both sides by 4, we find that a = 0.5. Now we can write the complete equation of the parabola as:

y = 0.5(x - 4)^2 - 3

We see that the correct answer must be option a, which matches the form of the equation we derived: (y = (x - 4)^2 - 3).