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Difficulty: EasyQuadratic Functions and Graphs

The graph of the quadratic function gg in the xyxy-plane is a parabola with vertex (2,−5)(2, -5). Which of the following equations could define the function gg?

  1. A
    g(x)=(x+2)2−5g(x) = (x + 2)^2 - 5
  2. g(x)=(x−2)2−5g(x) = (x - 2)^2 - 5Answer
  3. C
    g(x)=(x−2)2+5g(x) = (x - 2)^2 + 5
  4. D
    g(x)=(x+2)2+5g(x) = (x + 2)^2 + 5

Answer

The equation g(x)=(x−2)2−5g(x) = (x - 2)^2 - 5 could define the function gg.
The vertex form of a quadratic function is given by g(x)=a(x−h)2+kg(x) = a(x - h)^2 + k, where the point (h,k)(h, k) is the vertex of the parabola. We are given that the vertex is (2,−5)(2, -5), which means h=2h = 2 and k=−5k = -5. Substituting these values into the vertex form equation gives g(x)=a(x−2)2−5g(x) = a(x - 2)^2 - 5. Letting a=1a = 1 yields the equation g(x)=(x−2)2−5g(x) = (x - 2)^2 - 5.

Step-by-Step Solution

1
Identify the vertex form of a quadratic function.
The vertex form of a quadratic function is written as g(x)=a(x−h)2+kg(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola and aa is a non-zero constant.
Recognizing the vertex form allows us to directly substitute the coordinates of the given vertex.
2
Substitute the given vertex coordinates into the vertex form equation.
Given the vertex is (2,−5)(2, -5), we substitute h=2h = 2 and k=−5k = -5 into the vertex form: g(x)=a(x−2)2−5g(x) = a(x - 2)^2 - 5. If we assume a=1a = 1, this simplifies to g(x)=(x−2)2−5g(x) = (x - 2)^2 - 5.
This matches one of the given choices to find the equation that could define the function.

Key Concept

Vertex form of a quadratic function
Estimated Time:45s
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Topics

Quadratic Functions and Graphs

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