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Difficulty: EasyQuadratic Functions and Graphs

The quadratic function hh is defined by h(x)=(x+2)2−5h(x) = (x + 2)^2 - 5. What are the coordinates of the vertex of the graph of y=h(x)y = h(x) in the xyxy-plane?

  1. (−2,−5)(-2, -5)Answer
  2. B
    (2,−5)(2, -5)
  3. C
    (−2,5)(-2, 5)
  4. D
    (2,5)(2, 5)

Answer

The coordinates of the vertex of the graph are (−2,−5)(-2, -5).
The correct answer is the coordinate pair (−2,−5)(-2, -5). The vertex form of a quadratic function is given by y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. For the given equation h(x)=(x+2)2−5h(x) = (x + 2)^2 - 5, we rewrite the squared expression as (x−(−2))2(x - (-2))^2 to match the standard form. This shows that h=−2h = -2. The value of kk is the constant term −5-5. Therefore, the vertex of the graph of hh is (−2,−5)(-2, -5).

Step-by-Step Solution

1
Identify the general vertex form of a quadratic function.
The vertex form is y=a(x−h)2+ky = a(x - h)^2 + k, where the vertex of the parabola is at the point (h,k)(h, k).
This general equation allows us to directly read the coordinates of the vertex by comparing the coefficients.
2
Compare the given function h(x)=(x+2)2−5h(x) = (x + 2)^2 - 5 to the vertex form.
By matching the terms, we get x−h=x+2x - h = x + 2, which implies h=−2h = -2. The constant term is k=−5k = -5.
This identifies the values of hh and kk that make up the coordinates of the vertex (h,k)(h, k).
3
Write the vertex coordinates.
The vertex coordinates are (h,k)=(−2,−5)(h, k) = (-2, -5).
Combining the identified values gives the final coordinates of the vertex.

Key Concept

Identifying the vertex of a quadratic function directly from its vertex form equation, y=a(x−h)2+ky = a(x - h)^2 + k.
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Topics

Quadratic Functions and Graphs

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