Question

Gigi graphs the parabola (x+1)2=2(y−3).

Answer 1

After graphing the parabola $(x+1)^2=2(y-3)$, she can fill the boxes as follows:

Gigi identifies that the vertex of the parabola is $(-1, 3)$. The parabola opens up, and the focus is $\frac{1}{2}$ unit away from the vertex. The directrix is $1$ unit from the focus. The focus is the point $(-1, \frac{7}{2})$. The directrix of the equation is $y=\frac{5}{2}$.

Since Gigi's parabola has the x-term squared, then the parabola has the general formula

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

So,

$x-h=x+1 \implies h=-1$

$y-k=y-3\implies k=3$

The vertex is

$(h,k)=(-1,3)$

The focus is

$(h,k+p)=(-1,3+\frac{1}{2})\\=(-1,\frac{7}{2})$

The directrix is

$y=k-p=3-\frac{1}{2}=\frac{5}{2}$

The other values can be computed as follows;

Direction the parabola opens: Since the value of p is positive, the parabola opens up.

Distance of focus from vertex: The focus and vertex both have the same x position, so the distance will be in terms of the y-coordinates only. The distance from the focus to the vertex is

$\frac{7}{2}-3=\frac{1}{2}\text{ unit}$

Distance of directrix to focus: The directrix is the line $y=2.5$. The distance of the directrix to the focus will be

$\frac{7}{2}-\frac{5}{2}=1 \text{ unit}$

Learn more about conic sections here: brainly.com/question/12854078