Problem 26 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((5,-2) ;\) Focus: \((7,-2)\)

Step-by-Step Solution

Answer

The standard form equation of the parabola with vertex \((5,-2)\) and focus \((7,-2)\) is \((y+2)^2 = 8(x-5)\).

Step 1: Determine the direction of the parabola

In this step we figure out the direction of the parabola. The vertex of the parabola is \((5,-2)\) and the focus is \((7,-2)\). Noting the coordinates of the vertex and focus, the focus lies to the right of the vertex, hence the parabola opens to the right.

Step 2: Calculate the value of \(p\)

The value of \(p\) is the distance from the vertex to the focus point. We calculate the distance using their x-coordinates since the y-coordinates are the same, which simplifies the distance to the difference of the x-coordinates. The distance \(p = 7 - 5 = 2\).

Step 3: Write the equation in standard form

Now that we have the value of \(p\), we can fill the values into the standard form equation of a parabola. Our vertex is \((h, k) = (5,-2)\) and \(p = 2\). Substituting these into the standard form equation for a parabola that opens to the right gives: \((y-k)^2 = 4p(x-h)\) which simplifies to \((y+ 2)^2 = 8(x - 5)\).