Question

The vertex of the parabola \( 4y = x^2 - 6x + 17 \) is

• A. \( (3,2) \)
• B. \( (4,3) \)
• C. \( (4,2) \)
• D. \( (3,7) \)
• E. \( (7,2) \)

Hint:

Vertex form comes after completing square: \( y = a(x-h)^2 + k \Rightarrow (h,k) \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the vertex of a parabola, we need to rewrite its equation in the standard vertex form. For a parabola with a vertical axis of symmetry, the standard form is \((x - h)^2 = 4p(y - k)\), where (h, k) is the vertex.
Step 2: Key Formula or Approach:
The method involves completing the square for the terms involving `x`. 1. Rearrange the equation to group the x-terms. 2. Complete the square for the x-terms. 3. Factor the equation into the standard vertex form.
Step 3: Detailed Explanation:
The given equation is: \[ 4y = x^2 - 6x + 17 \] Isolate the x-terms on one side: \[ x^2 - 6x = 4y - 17 \] To complete the square for \(x^2 - 6x\), we take half of the coefficient of x, which is \(\frac{-6}{2} = -3\), and square it: \((-3)^2 = 9\). Add 9 to both sides of the equation: \[ (x^2 - 6x + 9) = 4y - 17 + 9 \] Factor the left side, which is now a perfect square trinomial, and simplify the right side: \[ (x - 3)^2 = 4y - 8 \] Factor out the coefficient of `y` on the right side to match the standard form: \[ (x - 3)^2 = 4(y - 2) \] Now, compare this with the standard form \((x - h)^2 = 4p(y - k)\). We can identify:
\(h = 3\)
\(k = 2\)
The vertex of the parabola is (h, k).
Step 4: Final Answer:
The vertex of the parabola is (3, 2).