The graph of a quadratic function is in the shape of a parabola. Parabolas representing quadratic functions open upward or downward.

There are many real-life situations where a parabolic shape occurs and can be modeled by a quadratic function. Some of these will be discussed in greater detail later in the Module.


The following tabs (1-9) are important information about quadratic functions and their graphs.

  • The equation of a quadratic function can be written in standard form as . This can also be represented in function notation as .



  • Quadratic functions are degree 2 polynomials. As such, the highest exponent on x (or another input variable) is 2 (). As a result, the equations of quadratic functions are relatively easy to identify when written in standard form.

  • The graphs of quadratic functions can open upward or downward. The sign of a in the equation of the function tells the direction of opening.

  • The graphs of quadratic functions are symmetrical about a vertical line, called the axis of symmetry, which passes through the centre of the parabola. As such, the left and right sides of the graph of a quadratic function are mirror images.

  • The graph of a quadratic function will always have either a maximum or a minimum value.

  • The maximum or minimum value of a quadratic function occurs at a point called the vertex. The axis of symmetry passes through the vertex and the x-coordinate of the vertex corresponds to the equation of the axis of symmetry.

The x-intercepts relate to the zeros of the graph of a quadratic function in the following way: when the value of the input (x) is an x-value that is located on the x-axis then the output value (f(x)) is a zero.

The x-intercepts relate to the roots of the graph of a quadratic equation in the same way. When y is equal to zero, then the x-value is located on the x-axis.

  • Quadratic functions may have 0, 1, or 2 zeros, just as the graphs of quadratic functions may have 0, 1, or 2 x-intercepts, as illustrated below.
  • For graphs of quadratic functions with two x-intercepts, the equation of the axis of symmetry can also be determined by finding the midpoint between the x-intercepts.

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