Question

find the standard equation of the parabola shown in the figure.

Explanation:

Step1: Recall parabola standard - form

The standard - form of a parabola with vertex \((h,k)\) is \(y=a(x - h)^2+k\). The vertex \(V(-2,-1)\), so \(h=-2\) and \(k = - 1\). Then the equation becomes \(y=a(x + 2)^2-1\).

Step2: Substitute the point \((2,1)\)

Substitute \(x = 2\) and \(y = 1\) into the equation \(y=a(x + 2)^2-1\). We get \(1=a(2 + 2)^2-1\).

Step3: Solve for \(a\)

First, simplify the equation \(1=a(4)^2-1\), which is \(1 = 16a-1\). Add 1 to both sides: \(1+1=16a\), so \(2 = 16a\). Then \(a=\frac{2}{16}=\frac{1}{8}\).

Step4: Write the final equation

Substitute \(a=\frac{1}{8}\) back into \(y=a(x + 2)^2-1\). The standard - equation of the parabola is \(y=\frac{1}{8}(x + 2)^2-1\).

Answer:

\(y=\frac{1}{8}(x + 2)^2-1\)