Find the equation, (f(x) = a(x - h)2 + k), for a parabola containing point (2, -1) and having (4, -3) as a vertex. What is the standard form of the equation?

Step-by-step explanation:

A parabola is written in the form

(1)

where:

is the x-coordinate of the vertex of the parabola

is the y-coordinate of the vertex of the parabola

is a scale factor

For the parabola in the problem, we know that the vertex has  coordinates (4,-3), so we have:

(2)

From this last equation, we get that (3)

Substituting (2) and (3) into (1) we get the new expression:

(4)

We also know that the parabola  contains the point (2,-1), so we can substitute

x = 2

f(x) = -1

Into eq.(4) and find the value of k:

So we also get:

So the equation of the parabola is:

(5)

Now we want to rewrite it in the standard form, i.e. in the form

To do that, we simply rewrite (5) expliciting the various terms, we find:

step-by-step explanation:

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