Problem 6 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}=4 x$$

Step-by-Step Solution

Answer

The focus of the parabola is at the point \((1,0)\), the directrix is the vertical line \(x=-1\).

Step 1: Identify the value of \(p\)

From the given equation of the parabola, \(y^{2}=4x\), we can rearrange the equation into the standard form \(y^{2}= 4px\). In this form, the coefficient of \(x\) is \(4p\). Therefore, we equate this to 4 as seen in the given equation to find the value of \(p\): \(4p = 4\). Solving for \(p\) gives: \(p = 1\)

Step 2: Find the focus of the parabola

For a parabola of the form \(y^{2}= 4px\), where \(p>0\), the focus is at the point \((p,0)\). Substituting \(p = 1\) into the coordinates, we get the focus at the point \((1,0)\).

Step 3: Find the directrix of the parabola

The directrix of a parabola of the form \(y^{2}= 4px\) is the vertical line \(x=-p\). Substituting \(p = 1\) into the expression gives the equation of the directrix as \(x=-1\).

Step 4: Graph the parabola

Plot the point \((1,0)\) which is the focus and draw the line \(x=-1\) which is the directrix. Sketch the parabola that opens to the right, and has its vertex at the origin (0,0), its focus at (1,0), and the directrix at \(x=-1\).