Question

1. write the equation of the parabola in vertex form that has a vertex of (3,-2) and passes through (5,-1). type a response

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Given vertex \((3, -2)\), so \(h = 3\), \(k=-2\). Substitute into the formula: \( y = a(x - 3)^2 - 2 \).

Step2: Substitute the point \((5, -1)\)

We know the parabola passes through \((5, -1)\), so substitute \(x = 5\), \(y = -1\) into \( y = a(x - 3)^2 - 2 \). We get \(-1 = a(5 - 3)^2 - 2\).

Step3: Solve for \(a\)

Simplify the equation: \(-1 = a(2)^2 - 2\) → \(-1 = 4a - 2\). Add 2 to both sides: \(4a = -1 + 2 = 1\). Then \(a=\frac{1}{4}\).

Step4: Write the final equation

Substitute \(a = \frac{1}{4}\), \(h = 3\), \(k = -2\) back into vertex form: \( y=\frac{1}{4}(x - 3)^2 - 2 \).

Answer:

\( y=\frac{1}{4}(x - 3)^2 - 2 \)