Question

A parabola has its vertex at (2, 2), and the equation of its directrix is y = 2.5. The equation that represents the parabola is

Answer 1

Answer:

In vertex form the equation is $y=-\frac{1}{2}(x-2)^2+2$

In standard form the equation is $y=-\frac{1}{2}x^2+2x$

Step-by-step explanation:

The equation of the directrix tells us that this is an x-squared parabola.  Because the directrix is above the vertex, the parabola will open downward.  The vertex form of this equation is:

$4p(y-k)=-(x-h)^2$

where p is the number of units between the directrix and the vertex.  The number of units here is .5 or 1/2.  Filling in the coordinates of the vertex and the p value of 1/2:

$4(.5)(y-2)=-(x-2)^2$

Simplifying we have:

$2(y-2)=-(x-2)^2$

Divide both sides by 2 to get:

$y-2=-\frac{1}{2}(x-2)^2$

Add 2 to both sides to get the final vertex form:

$y=-\frac{1}{2}(x-2)^2+2$

If you want that in standard form, you first need to expand the squared term to get:

$y=-\frac{1}{2}(x^2-4x+4)+2$

Order of operations tells us that we have to distribute in the -1/2 first to get:

$y=-\frac{1}{2}x^2+2x+2-2$

which simplifies to the standard form:

$y=-\frac{1}{2}x^2+2x$