Planning tool

Students:

• use digital tools to manipulate, explore, investigate, make conjectures and identify emerging patterns from functions and relations, for example, transformations of the graph of the unit circle
• explore linear, quadratic and exponential functions and related algebra, tables of values, number patterns and graphs involving non-integer roots
• use digital tools to zoom in on and progressively refine approximations of points of intersection between two functions
• apply a bisection method to locate approximate values for the x-intercepts of quadratic functions with irrational roots, such as
  f(x) = 3x² – 4x – 6.

Some students may:

• not realise that irrational numbers are infinite when expressed as decimal numbers and hence are only represented by an approximation of their exact value.

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 

• We are learning to use the zoom-in functionality of graphing software to find the approximate location of points of intersection of two graphs.

Why are we learning about this? 

Many practical problems involve solving equations that do not have exact solutions, or where the process of finding the exact solution is complicated and an approximation is sufficient. For example, many calculators and graphing software use algorithms to calculate approximate answers of irrational numbers when expressed as a decimal.

What to do 

Intersecting graphs

Use a graphing tool such as a graphics calculator or GeoGebra to complete the following.

1. Graph the functions y = 2x and y = x² on the same set of axes and find the coordinates of any points of intersection of the graphs.
2. Graph the functions y = 2x and y = 2^x on the same set of axes and find the coordinates of any points of intersection of the graphs.
3. Graph the functions y = x² and y = 2^x on the same set of axes and explore their points of intersection:
   1. Find the coordinates of the two points of intersection that have positive integer x and y.
   2. Find another point of intersection that is between –1 and 0, and has a decimal y value between 0 and 1. Find approximate coordinates for this point correct to 1 decimal place.
   3. Use the graphing tool ‘zoom’ feature to increase the detail around the point of intersection (more intervals with smaller decimal differences) and show that the x value of the third intercept is between –0.80 and –0.75. Find approximate coordinates for this point, correct to 2 decimal places.
   4. Continue zooming in on the point of intersection until the graphs almost look like two straight lines crossing. Then find approximate coordinates for the point of intersection correct to 3 decimal places.
4. Use this approach to find approximate coordinates for the points of intersection of the functions y = 3x and y = 3^x, correct to 3 decimal places.

Success criteria 

• I can find the approximate location of any points of intersection of the graphs of two functions and systematically refine these approximations using the zoom functionality of graphing software.