Question

select the correct answer.
what is the equation of the parabola shown with its directrix on this graph?
a. $y = \frac{1}{8}(x - 1)^2 - 2$
b. $y = \frac{1}{4}(x - 1)^2 - 2$
c. $y = 8(x - 1)^2 - 2$
d. $y = 4(x - 1)^2 - 2$

Explanation:

Step1: Identify vertex of parabola

From the graph, the vertex $(h,k)$ is $(1, -2)$.

Step2: Recall parabola vertex form

The vertex form is $y = a(x-h)^2 + k$, so substitute $h=1, k=-2$:
$y = a(x-1)^2 - 2$

Step3: Find directrix and calculate $a$

The directrix is the dashed line $y=-3$. For a vertical parabola, the distance from vertex to directrix is $|p| = |-2 - (-3)| = 1$. The relationship is $a = \frac{1}{4p}$.
Substitute $p=1$:
$a = \frac{1}{4(1)} = \frac{1}{4}$

Step4: Write final equation

Substitute $a=\frac{1}{4}$ into the vertex form:
$y = \frac{1}{4}(x - 1)^2 - 2$

Answer:

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