Question

in the vertex form of a quadratic equation, $y = a(x - h)^2 + k$, how is the vertex represented?
a. $(k, a)$
b. $(h, k)$
c. $(a, h)$
d. $(a, k)$

Explanation:

Step1: Recall Vertex Form Definition

The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where the vertex of the parabola (the graph of the quadratic function) is at the point \((h, k)\). This is because the term \((x - h)\) determines the horizontal shift (with \(h\) being the x - coordinate of the vertex) and \(k\) is the vertical shift (the y - coordinate of the vertex).

Step2: Analyze the Options

• Option a: \((k,a)\) does not match the vertex form definition. \(k\) is the y - coordinate and \(a\) is the coefficient that determines the direction and width of the parabola, not the vertex coordinates.
• Option b: \((h,k)\) matches the definition of the vertex in the vertex form \(y=a(x - h)^2 + k\).
• Option c: \((a,h)\) is incorrect. \(a\) is not a coordinate of the vertex, and \(h\) is the x - coordinate, but the order and the first coordinate are wrong.
• Option d: \((a,k)\) is incorrect. \(a\) is not a coordinate of the vertex.

Answer:

b. \((h,k)\)