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Difficulty: EasyQuadratic Functions and Graphs

The function ff is defined by f(x)=(x−5)2+3f(x) = (x - 5)^2 + 3. If the graph of y=f(x)y = f(x) is translated 4 units down in the xyxy-plane to create the graph of the function gg, what is the vertex of the graph of gg?

  1. A
    (1,3)(1, 3)
  2. B
    (−5,−1)(-5, -1)
  3. (5,−1)(5, -1)Answer
  4. D
    (5,7)(5, 7)

Answer

The vertex of the graph of gg is (5,−1)(5, -1).
The vertex of a quadratic function of the form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k is (h,k)(h, k). For the function f(x)=(x−5)2+3f(x) = (x - 5)^2 + 3, the vertex is (5,3)(5, 3). Translating the graph of a function vertically downward by 4 units decreases the yy-coordinate of every point on the graph by 4, while the xx-coordinate remains unchanged. Applying this transformation to the vertex coordinates yields (5,3−4)(5, 3 - 4), which simplifies to (5,−1)(5, -1).

Step-by-Step Solution

1
Identify the vertex of the original function f(x)=(x−5)2+3f(x) = (x - 5)^2 + 3.
The vertex of the parabola y=a(x−h)2+ky = a(x - h)^2 + k is (h,k)(h, k). For f(x)=(x−5)2+3f(x) = (x - 5)^2 + 3, the vertex is (5,3)(5, 3).
Finding the original vertex is the starting point for applying the translation.
2
Apply the vertical translation of 4 units downward to the vertex coordinates.
A downward translation of 4 units subtracts 4 from the yy-coordinate of the vertex: (5,3−4)=(5,−1)(5, 3 - 4) = (5, -1).
Translating a graph vertically downward decreases the yy-value of all its points by the translation distance.

Key Concept

Identifying the vertex of a quadratic function in vertex form and applying vertical translations.
Estimated Time:45s
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Topics

Quadratic Functions and Graphs

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