Embedded Files
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:
- The general (polynomial) form: y = ax2 + bx + c
- The turning point form: y = a(x - h)2 + k
- 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 = ax2.
- 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 = ax2.
Figure 1 - The effect of a on the graph y = ax2.
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 = x2 + 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 = x2 + k.
Figure 2 - The effect of k on the graph y = x2 + 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:
- Determine the y-intercept of a quadratic function.
- Determine the x-intercepts of a quadratic function.
- Determine the turning point of a quadratic function.
- Plot the axes intercepts and turning point on a set of axes and sketch the parabola.
- Label all key features of a quadratic with their coordinates.
Page updated
Google Sites
Report abuse