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Chapter 3: Problem 97

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

Short Answer

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

Step by step solution

01

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.
02

Substitute the vertex into the equation

Substitute the vertex into both forms of the equations, yielding \(y = a(x+3)^2 - 4\) and \(x = a(y+4)^2 - 3\).
03

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 \(y = a(x+3)^2 - 4\).
04

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: \(y = (x+3)^2 - 4\).

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