Problem 13 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}-6 x=0$$

Step-by-Step Solution

Answer

The focus of the given parabola is at point (1.5, 0) and the equation of the directrix is x = -1.5. The graph of the parabola opens to the right, passing through the focus and is parallel to the directrix.

Step 1: Rewrite Equation in Standard Form

Reframing the given equation \(y^2 - 6x = 0 \) in standard form to obtain: \(y^2=6x\)

Step 2: Identify the form and value of 4p

Identify if the equation is in the form \(y^2 = 4px\) or \(x^2 = 4py\). Here, equation is in the form \(y^2 = 4px \) where \(4p = 6\) and hence, \(p = 1.5\). The value of p helps us in calculating the focus and directrix.

Step 3: Determine focus and directrix

For the parabola in the form \(y^2 = 4px\), the focus is at (p,0) and directrix is at x = -p. So, for the given equation, the focus is at (1.5, 0) and directrix is at x = -1.5.

Step 4: Graph the parabola

Plot the focus point on the x-axis and draw the directrix line at x = -1.5. Lastly, plot the parabola which will open towards the right (as the equation is in the form \(y^2 = 4px\)) and passing through the focus and parallel to the directrix.