Planning tool

Students:

• vary parameters in rules of functions
• use digital tools to develop and manipulate a broad range of examples
• apply transformations in the coordinate plane to graphs of linear and quadratic functions
• identify patterns resulting from systematic use of transformations on the graphs of other functions, such as reciprocal, square and square root representations, when parameters are varied in their rules.

Some students may:

• have difficulty connecting geometric transformations of shapes on a plane surface to transformations of graphs of functions in the coordinate plane.
• have difficulty understanding that the sign of the term in (x + h)² is opposite to the direction of horizontal translation,
  for example (x + 3)² corresponds to a horizontal shift of 3 units to the left.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

• I am learning to connect rules of quadratic functions with the shape of the corresponding graphs.
• I am learning to identify patterns resulting from systematically varying the parameters of the quadratic function y = a(x + h)² + k.

Why are we learning about this?

The graphs of quadratic functions, which are called parabolas, can be described in terms of patterns based on shifts, stretches and reflections of the graph of the basic parabola y = x². This makes it easier to understand the connection between the rules of quadratic functions in vertex form, such as y = 2(x – 3)² + 4, and the shape of their graph.

What to do

Use the GeoGebra Parabolas: graphing vertex form to draw graphs of quadratics and observe patterns resulting from changing numbers in the rule.

Shifting the parabola around

1. To start, open the graphing software and type y = a(x + h)² + k. Note the sliders for each of the variables a, h and k.
2. Keep a = 1 and use the sliders to make h = 0 and k = 0. The graph of the basic parabola y = x² will be shown, with its vertex at the point (0, 0).
3. Keep a = 1 and h = 0 and change the value of k in steps of 1 from 0 to –5. Describe what happens to the graphs shown in terms of the value of k.
4. Now have a = 1 and k = 0 and change the value of h in steps of 1 from 0 to –5. Describe what happens to the graphs shown in terms of the value of h.
5. Find the values of h and k, with a = 1, for which the vertex of the parabola is at the following points: (2, 3), and (–4, 5).

Stretching and flipping the parabola

1. Make h = 0 and k = 0 and change the value of a in steps of 0.5 from 0.5 to 4. Describe what happens to the graphs shown in terms of the value of a.
2. Keep h = 0 and k = 0 and change the value of a in steps of 0.5 from –0.5 to –4. Describe what happens to the graphs shown in terms of the value of a.

Using rules to draw graphs and fit points

1. Use the sliders to draw the graphs of:
   
   y = 0.5(x + 4)² – 7 and y = –2(x – 1)² + 16
   
   
2. Find the values of a, h and k, for which the graph of the parabola passes through the points with coordinates
   (–2, 0), (0, 20) and (2, 0).

Success criteria

• I can connect rules of quadratic functions with the shape of the corresponding graphs.
• I can identify patterns resulting from systematically varying the parameters of the quadratic function y = a(x + h)² + k.