Problem 97 - College Algebra

Question

Write the equation of each parabola in standard form. Vertex: \((-3,-4) ;\) The graph passes through the point \((1,4)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y = (x+3)^2 - 4\).

Step 1: Identify the vertex

The vertex of the parabola is given as \(-3, -4\). This will be substituted into the standard form equation. It's not specified whether the parabola is facing upwards, downwards, or sideways, so both options will be considered.

Step 2: Substitute the vertex into the equation

Substitute the vertex into both forms of the equations, yielding \(y = a(x+3)^2 - 4\) and \(x = a(y+4)^2 - 3\).

Step 3: Substitute the other point into the equation

Substitute the given point \((1,4)\) into both of our possible equations. Starting with the first equation \(y = a(x+3)^2 - 4\), when \(x = 1\) and \(y = 4\), you get \(4 = a(1+3)^2 - 4\). Solving for \(a\) results in \(a=1\). When substituting into the other equation, \(x = a(y+4)^2 - 3\), this leads to fractional \(a\) which is not as plausible. Thus the equation is probably \(y = a(x+3)^2 - 4\).

Step 4: Write the final equation

Now all of the coefficients in the standard equation are known. Recall that \(a = 1\), so substitute that into the equation \(y = a(x+3)^2 - 4\). This gives the final equation of the parabola: \(y = (x+3)^2 - 4\).