### Cubic Spin

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

### Sine Problem

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

### More Parabolic Patterns

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

# Parabolic Patterns

### Why dothis problem?

In this problem, instead of giving the equations of some functions and asking students to sketch the graphs, this challenge gives the graphs and asks them to find their equations. This encourages students to experiment by changing the equations systematically to discover the effect on the graphs.

### Possible approach

This worksheet might be useful.

Start by showing this picture and ask students to work in pairs to identify the graphs of $y=x^2$ and $y=-(x-4)^2$.
Ask them to describe what is the same and what is different about the two curves.
Explain: "Your challenge is to recreate the whole pattern using graphing software. You need to work out how changing the equation affects the shape and position of the graph."

Give students plenty of time to experiment with the graphing software.
After a while, bring the class together and share any useful insights they have noticed. Here are some examples of the sort of observations students may make:
• $y = −x^2$ is the same as $y = x^2$ but flipped upside down.
• $y = x^2 + 4$ is the same as $y = x^2$ but moved up four spaces.
• $y = (x − 3)^2$ is the same as $y = x^2$ but moved right three spaces.
After sharing their observations, give students a little longer to use these new ideas to complete the original task. Then challenge them to create a pattern of their own using parabolas.

To finish off, students could print out their creations, give their printout to another group, and challenge them to work out which equations were used to create it.

### Key questions

• You are being asked to sketch a family of graphs. What makes this a family?
• What is the same and what is different about the equations $y=x^2$ and $y=-(x-4)^2$?
• How might these similarities and differences relate to the way they look and their positions on the axes?
• Can you convince us that the rules you have found will work with graphs of other functions.

### Possible extension

This worksheet contains a second set of graphs for students to identify, which focusses on stretches.

More Parabolic Patterns and Parabolas again offer similar pictures to reproduce.

Cubics uses graphs of cubic functions, and Ellipses gives the opportunity to investigate the equation of an ellipse.

### Possible Support

Students could begin by investigating translation of straight lines and look at how their equations change.