Question

match the equation with the focus, vertex, directrix, and axis of symmetry. use the blank graph on the next question if you want to sketch the graph to help you determine the attributes.

equation	vertex	focus	axis of symmetry	directrix

(-4, -1) (5, 3) (0, 4) (3, 5) (1, 3) (-1, -4) (-2, -1) (-1, -3) y = -1
x = 0 y = 0 x = -3 x = -1 y = 3 y = -3

Explanation:

Step1: Recall formula for parabola

The equation of a parabola of the form $x = a(y - k)^2+h$ has vertex $(h,k)$. Here $h=-2,k = - 1,a=-\frac{1}{8}$. The focus - vertex relationship for a parabola $x=a(y - k)^2+h$ is given by the formula $(h +\frac{1}{4a},k)$.

Step2: Calculate the focus

Substitute $h=-2,a =-\frac{1}{8},k=-1$ into the focus formula. First, find $\frac{1}{4a}=\frac{1}{4\times(-\frac{1}{8})}=- 2$. Then $h+\frac{1}{4a}=-2-2=-4$. So the focus is $(-4,-1)$.

Step3: Determine the axis of symmetry

For a parabola of the form $x=a(y - k)^2+h$, the axis of symmetry is the line $y = k$. Since $k=-1$, the axis of symmetry is $y=-1$.

Step4: Find the directrix

The directrix of a parabola $x=a(y - k)^2+h$ is given by the equation $x=h-\frac{1}{4a}$. We know $h = - 2$ and $\frac{1}{4a}=-2$, so $x=-2-(-2)=-3$.

Answer:

Focus: $(-4,-1)$
Axis of Symmetry: $y=-1$
Directrix: $x=-3$