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Difficulty: MediumQuadratic Functions and Graphs

The graph of the quadratic function ff in the xyxy-plane is a parabola. If f(−3)=−2f(-3) = -2 and f(7)=−2f(7) = -2, what is the xx-coordinate of the vertex of the parabola?

  1. A
    4
  2. 2Answer
  3. C
    -2
  4. D
    5

Answer

The correct answer is 2, since the vertex lies on the axis of symmetry, which is the midpoint of the symmetric x-coordinates.
The vertex of a parabola in the xyxy-plane lies on its axis of symmetry. For any quadratic function, points with the same yy-coordinate are symmetric with respect to the axis of symmetry. Since f(−3)=−2f(-3) = -2 and f(7)=−2f(7) = -2, the yy-coordinates are equal, which means the axis of symmetry is the vertical line midway between x=−3x = -3 and x=7x = 7. The xx-coordinate of the vertex is the midpoint of these two xx-values, calculated as −3+72=2\frac{-3 + 7}{2} = 2.

Step-by-Step Solution

1
Identify the two points on the parabola with the same y-value.
The points are (−3,−2)(-3, -2) and (7,−2)(7, -2).
Since both points share the same y-value of −2-2, they are symmetric with respect to the axis of symmetry of the parabola.
2
Find the x-coordinate of the axis of symmetry by calculating the midpoint of the x-coordinates of the two symmetric points.
Midpoint=−3+72=42=2\text{Midpoint} = \frac{-3 + 7}{2} = \frac{4}{2} = 2.
The axis of symmetry is always located exactly halfway between any two symmetric points on the parabola.
3
Determine the x-coordinate of the vertex of the parabola.
The x-coordinate of the vertex is 2.
The vertex of a parabola lies on its axis of symmetry, so its x-coordinate is equal to the x-coordinate of the axis of symmetry.

Key Concept

Symmetry of quadratic graphs and the axis of symmetry

Alternative Method

Alternatively, any quadratic function that takes the value −2-2 at x=−3x = -3 and x=7x = 7 can be written in the form f(x)=a(x+3)(x−7)−2f(x) = a(x + 3)(x - 7) - 2 for some constant aa. Expanding this expression yields f(x)=a(x2−4x−21)−2f(x) = a(x^2 - 4x - 21) - 2. Since the xx-coordinate of the vertex of a quadratic function in standard form y=ax2+bx+cy = ax^2 + bx + c is given by −b2a-\frac{b}{2a}, we can find the vertex xx-coordinate as −−4a2a=2-\frac{-4a}{2a} = 2.
Estimated Time:1m 30s
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Topics

Quadratic Functions and Graphs

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