4G - Graphing quadratic functions

Learning intentions:

In this section we will examine:

    • Transformations of the quadratic function in the form y = a(x - h)2 + k.
    • Graphing quadratic functions from any form (general, factorised or turning point).
    • Labelling key features of a parabola.

Transformations of the quadratic function

A quadratic function can exist in three forms:

    1. The general (polynomial) form: y = ax2 + bx + c
    2. The turning point form: y = a(x - h)2 + k
    3. Factorised form: y = a(x - m)(x - n)

The most useful for is the turning point form when we discuss transformations. Consider the turning point form:

We can consider the effects of each parameter (a, h and k) on the graph of the parabola.

    • a will cause a dilation by a factor of a from the x-axis.
    • h will cause a horizontal translation of h units.
    • k will cause a vertical translation of k units.

Before discussing the transformations of a quadratic, always make sure it is in turning point form. If it is not in turning point form, complete the square on the general form.

Examining the individual effects of a, h and k

The effects of the parameter a

The dynamic GeoGebra worksheet illustrates the effect of a on the graph of y = ax 2.

    • Please click on the play button in the bottom left hand corner to animate!

The graph below show the effect of a on the graph of y = ax 2.

Figure 1 - The effect of a on the graph y = ax 2.

From the graph above we can see that:

    • When a > 0 (positive) the parabola is concave up.
      • If 0 < a < 1 the parabola is concave up and rises slowly (graph is broader).
      • If a > 1 the graph is concave up and rises steeply (graph is narrower).
    • When a < 0 (negative) the parabola is concave down.
    • If - 1 < a < 0 the parabola is concave down and falls slowly (graph is broader).
      • If a < - 1 the parabola concave down and falls steeply (graph is narrower).

a causes a dilation by a factor of a from the x-axis.

The effects of the parameter h

The dynamic GeoGebra worksheet illustrates the effect of h on the graph of y = (x - h)2.

    • Please click on the play button in the bottom left hand corner to animate!

The graph below show the effect of h on the graph of y = (x - h)2.

Figure 2 - The effect of h on the graph y = (x - h)2.

From the graph above we can see that:

    • When h > 0 (positive) the parabola translated h units in the negative x-direction.
    • When h < 0 (negative) the parabola translated h units in the positive x-direction.

The effects of the parameter k

The dynamic GeoGebra worksheet illustrates the effect of k on the graph of y = x 2 + k.

    • Please click on the play button in the bottom left hand corner to animate!

The graph below show the effect of k on the graph of y = x 2 + k.

Figure 2 - The effect of k on the graph y = x 2 + k.

From the graph above we can see that:

    • When k > 0 (positive) the parabola translated h units in the positive y-direction.
    • When k < 0 (negative) the parabola translated h units in the negative y-direction.

Examining the combined effects of a, h and k

The dynamic GeoGebra worksheet illustrates the combined effect of a, h and k on the graph of y = a(x + h)2 + k.

    • Please use the sliders to adjust the parameters and observe the transformations.

Graphing quadratic functions

When graphing quadratics (parabolas), we need to show clearly:

    • x-intercepts (if there are any).
    • y-intercepts.
    • The turning point.

Figure 1 - The graph of y = x2 + 2x - 8 with key features labelled.

Method: Graphing quadratics

Step 1: Find the y-intercept

    • If the equation is in the general from (ax2 + bx + c) then the y-intercept is (0, c).
    • If the equation is not in the general form, let x = 0 and solve for y.

Step 2: Find the x-intercept

    • The easiest way to find the x-intercepts is to get the quadratic into factorised form and let y = 0. The null factor law can then be used to solve the quadratic equation for x.
  • If the equation cannot be easily factorised, then the general quadratic formula can be used to find the x-intercepts. When the equation is in the general form (ax2 + bx + c), the general quadratic formula states:

    • Remember: you can use the discriminant (Δ) to determine how many x-intercepts exist:

Step 3: Find the turning point

    • The easiest way to find the turning point is when the quadratic is in turning point form (y = a(x - h) 2 + k), where (h, k) is the turning point. To get a quadratic into turning point form you need to complete the square.
    • Otherwise, you can use the axis of symmetry to determine the x-coordinate of the turning point, the y-coordinate can be found by substituting the x-coordinate into the quadratic equation. The axis of symmetry is given by:

    • Finally, you can use differential calculus to determine the stationary point of the quadratic which is the turning point of a parabola.

Step 4: Draw a parabola through points from (1) - (3) on a set of axes

    • Locate each of the points found in step (1), (2) and (3) on a set of axes.
    • Determine if the parabola is positive or negative from the value of a in the general form

Figure 2 - The general shape of a positive and negative parabola.

  • The three points plotted from step (1), (2) and (3) should guide you to the overall shape of the parabola; however, it is always a good idea to know what shape you should expect.

Step 5: Label all points with their coordinates

Lastly, and most importantly, ensure all the important features of the graph have been labelled with their coordinates. Remember the important features are:

    • x-intercepts (if there are any).
    • y-intercepts.
    • The turning point.

4G - VIDEO EXAMPLE 1:

Graph the following quadratic equation:

4G - VIDEO EXAMPLE 2:

Graph the following quadratic equation:

4G - VIDEO EXAMPLE 3:

Graph the following quadratic equation:

Success criteria:

You will be successful if you can:

    1. Determine the y -intercept of a quadratic function.
    2. Determine the x -intercepts of a quadratic function.
    3. Determine the turning point of a quadratic function.
    4. Plot the axes intercepts and turning point on a set of axes and sketch the parabola.
    5. Label all key features of a quadratic with their coordinates.