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How do you represent a quadratic function graphically?

Published in Quadratic Functions 3 mins read

A quadratic function is represented graphically as a parabola, which is a U-shaped curve.

Here's a breakdown:

  • What is a Parabola? A parabola is a symmetrical, curved shape. All quadratic functions, when graphed on a coordinate plane, produce a parabola.

  • Equation Form: A quadratic function is typically written in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The value of 'a' dictates whether the parabola opens upwards or downwards.

  • Opening Up or Down:

    • If a > 0 (a is positive), the parabola opens upwards, resembling a smiling face (concave up).
    • If a < 0 (a is negative), the parabola opens downwards, resembling a frowning face (concave down).

  • Vertex: The vertex is the turning point of the parabola. It's either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). The x-coordinate of the vertex can be found using the formula x = -b / 2a. You can then substitute this x-value back into the quadratic function to find the y-coordinate of the vertex.

  • Axis of Symmetry: This is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. Its equation is x = -b / 2a.

  • X-intercepts (Roots/Zeros): These are the points where the parabola intersects the x-axis. To find them, set f(x) = 0 and solve for x. You can use factoring, the quadratic formula, or completing the square.

  • Y-intercept: This is the point where the parabola intersects the y-axis. To find it, set x = 0 in the quadratic function. The y-intercept is always at the point (0, c).

Example:

Consider the quadratic function f(x) = x² - 4x + 3.

  1. Shape: Since a = 1 (which is positive), the parabola opens upwards.
  2. Vertex: x = -(-4) / (2 * 1) = 2. f(2) = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1. Therefore, the vertex is at (2, -1).
  3. Axis of Symmetry: x = 2.
  4. X-intercepts: Set x² - 4x + 3 = 0. Factoring gives (x - 3)(x - 1) = 0. So, the x-intercepts are x = 1 and x = 3, or the points (1, 0) and (3, 0).
  5. Y-intercept: Set x = 0. f(0) = (0)² - 4(0) + 3 = 3. So, the y-intercept is at (0, 3).

By plotting these points (vertex, intercepts) and knowing the shape (opening upwards), you can accurately sketch the parabola.

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