Problem 5 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}=16 x$$

Step-by-Step Solution

Answer

The focus of the given parabola is at (4,0) and the equation of the directrix is \( x=-4 \).

Step 1: Identify standard form parameters

The given equation is \( y^2=16x \). Compare this with the standard equation \( y^2=4px \). From comparison, 4p=16, we get p=4.

Step 2: Find the Focus

For a parabola of the form \( y^2=4px \), the focus is at (p,0). Hence, using the obtained p value, the focus of the parabola is (4, 0).

Step 3: Find the Directrix

For a parabola of the form \( y^2=4px \), the directrix is given by the line \( x=-p \). Hence, the directrix for this parabola is \( x=-4 \).

Step 4: Graph the Parabola

Now, plot the parabola using the obtained focus and directrix. The parabola facing to the right with the vertex at the origin, focus at (4, 0) and directrix at \( x=-4 \).