Question

find the focus and directrix of the parabola with the given equation. then graph the parabola.
$y^2 = 8x$

what is the focus?
(type an ordered pair. simplify your answer.)

what is the directrix?
(type an equation. simplify your answer.)

choose the correct graph of the parabola.
a. graph of parabola opening upwards
b. graph of parabola opening to the right
c. partial graph of parabola

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens to the right or left is \( y^2 = 4px \), where the vertex is at \((0,0)\), the focus is at \((p,0)\), and the directrix is the line \( x=-p \).

Step2: Compare the given equation with the standard form

Given the equation \( y^2 = 8x \). Comparing with \( y^2 = 4px \), we have \( 4p = 8 \).

Step3: Solve for \( p \)

To find \( p \), divide both sides of \( 4p = 8 \) by 4: \( p=\frac{8}{4}=2 \).

Step4: Find the focus

Since the focus of \( y^2 = 4px \) is \((p,0)\) and \( p = 2 \), the focus is \((2,0)\).

Step5: Find the directrix

The directrix of \( y^2 = 4px \) is \( x=-p \). Substituting \( p = 2 \), the directrix is \( x=-2 \).

Step6: Identify the correct graph

The parabola \( y^2 = 8x \) opens to the right (because the coefficient of \( x \) is positive and \( y \) is squared). So the graph should open to the right, which matches option B.

Answer:

Focus: \((2, 0)\)
Directrix: \( x = -2 \)
Graph: B