Problem 8 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}=-12 x$$

Step-by-Step Solution

Answer

The focus of the given parabola is at the point \(-3, 0\) and the equation of directrix is \(x = 3\).

Step 1: Convert the equation into standard form

To convert the given equation into standard form, it has to be written as \(4p(x-h)= (y-k)^{2}\) for horizontal parabolas. We have \(y^{2}= -12x\), so it will be written as \(-4(3x)= y^{2}\). Comparing it to the standard form, we can say that \(h=0, k=0, p=-3\).

Step 2: Finding the focus

For horizontal parabolas, the focus is calculated using the formula \(h+p, k\), substituting the values obtained from our previous step, our focus will be \(-3,0\).

Step 3: Finding Directrix

For horizontal parabolas, the directrix is calculated using the formula \(x=h-p\). Substituting the obtained values, we get our directrix as \(x=3\).

Step 4: Graph the Parabola

To graph the parabola, we need to firstly plot the vertex which is the point (h,k) obtained in step 1. Additionally, we plot the focus found in step 2. Also, draw a vertical line corresponding to the directrix equation obtained in step 3. Now, draw a smooth, continuous curve representing the parabola that touches the vertex, outlines the focus inside and is tangent to the directrix.