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Difficulty: MediumQuadratic Functions and Graphs

The graph of the quadratic function ff in the xyxy-plane has its vertex at (2,−5)(2, -5). If the graph passes through the point (5,13)(5, 13), what is the value of f(−1)f(-1)?

  1. A
    -23
  2. B
    -3
  3. 13Answer
  4. D
    -11

Answer

13
The vertex of the parabola is (2,−5)(2, -5), which indicates that the axis of symmetry is the vertical line x=2x = 2. The given point has an xx-coordinate of 55, which is a distance of 33 units from the axis of symmetry (5−2=35 - 2 = 3). The target point has an xx-coordinate of −1-1, which is also a distance of 33 units from the axis of symmetry (2−(−1)=32 - (-1) = 3). Because a parabola is perfectly symmetric about its axis of symmetry, any two points that are the same horizontal distance from this line must have the same yy-coordinate. Thus, f(−1)f(-1) must be equal to f(5)f(5), which is 1313. Alternatively, one can find the specific equation of the quadratic function by substituting the vertex and the point (5,13)(5, 13) into the vertex form f(x)=a(x−2)2−5f(x) = a(x - 2)^2 - 5, yielding a=2a = 2. Evaluating f(−1)=2(−1−2)2−5f(-1) = 2(-1 - 2)^2 - 5 gives 1313.

Step-by-Step Solution

1
Identify the axis of symmetry from the given vertex.
The axis of symmetry is x=2x = 2.
For any quadratic function with a vertex at (h,k)(h, k), the vertical line x=hx = h is the axis of symmetry of its parabolic graph.
2
Determine the horizontal distance from the axis of symmetry to the given point x=5x = 5 and the target point x=−1x = -1.
The distance for x=5x = 5 is 5−2=35 - 2 = 3 units. The distance for x=−1x = -1 is 2−(−1)=32 - (-1) = 3 units.
Checking if the two xx-coordinates are symmetric with respect to the line x=2x = 2 allows us to use the symmetry property of parabolas.
3
Apply the symmetry property to find the function value.
Since both x=5x = 5 and x=−1x = -1 are equidistant from the axis of symmetry, their function values are equal: f(−1)=f(5)=13f(-1) = f(5) = 13.
Points on a parabola that are equidistant from the axis of symmetry have the same yy-coordinate.

Key Concept

Symmetry of quadratic functions about their vertex axis of symmetry
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Topics

Quadratic Functions and Graphs

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