Parabolic Patterns
The illustration shows the graphs of fifteen functions. Two of them have equations $y=x^2$ and $y=-(x-4)^2$. Find the equations of all the other graphs.
Problem
Parabolic Patterns printable sheet
The illustration shows the graphs of fifteen functions. Two of them have equations
$y = x^2$
$y = - (x - 4)^2$
Can you find the equations of the other parabolas in the picture?
You may wish to use a graphical calculator or software such as Desmos to recreate the pattern for yourself.
Can you find the equations of these parabolas?
NOTES AND BACKGROUND
This sort of challenge is sometimes called an inverse problem because the question is posed the opposite way round to what might have been expected. This is almost like saying: 'here is the answer, what was the question?' Instead of giving the equations of some functions and asking you to sketch the graphs, this challenge gives the graphs and asks you to find their equations.
You are being asked to sketch a family of graphs. What makes this a family? All the graphs are obtained by transformations such as reflections and translations of other graphs in the family. The key is to find the simplest function and then to find transformations of the graph of that function which give the other graphs in the family.
If you have access to a graphic calculator, or to graph drawing software, it will not give you the answers. You will have to think for yourself what the equations should be and then the software will enable you to test your own theories and see if you were right.
Getting Started
Try sketching the graph of $y=x^2$ on paper. What would you expect the graph of $y=-x^2$ to look like? What is the effect of the minus sign? Is this one of the graphs in the picture?
What would you expect the graph of $y=(x-4)^2$ to look like? How would you expect the graph of $y=x^2$ to be transformed to give the graph of $y=(x-4)^2$?
What about $y=-(x-4)^2$?
Draw the graphs of these functions using graph drawing software or a graphics calculator if you have access to one or the other. Were your predictions right?
What have you learnt from this example about reflections and translations of graphs and the corresponding equations of the functions?
Now experiment with drawing the graphs of other functions and see if you can find the equations for all the graphs in the picture.
Student Solutions
We have had solutions from Fiona, a Year 10 student from Stamford High School (Lincolnshire), and from Bei Guo, Kristin and Ryan from Riccarton High School in Christchurch (New Zealand). Well done to you all
They all noticed that the parabolas could be treated as three separate sets of five parabolas.
Starting with $y = x^2$, the parabola going through the origin, they noticed that it could be raised or lowered to produce the rest of the parabolas in the middle set by either adding or subtracting a $2$ or a $4$.
Therefore the solutions for the middle set are:
$y = x^2$
$y = x^2 + 2$
$y = x^2 + 4$
$y = x^2 - 2$
$y = x^2 - 4$
Then they used the other equation that had been given, $y = - (x - 4)^2$ , and found that it gave one of the parabolas in the right hand set.
$y = - (x^2 )$ is a reflection of $y = x^2$ in the horizontal axis, so that is why the new parabola was an inverted version of the original one.
$y = (x - 4)^2$ is a translation of $y = x^2$ by $4$ units to the right, so that is why the new parabola was the inverted parabola shifted $4$ units to the right.
As before, they noticed that $y = - (x - 4)^2$ could be raised or lowered to produce the rest of the parabolas in the right hand set by either adding or subtracting a $2$ or a $4$.
Therefore the solutions for the right hand set are:
$y = - (x - 4)^2$
$y = - (x - 4)^2 + 2$
$y = - (x - 4)^2 + 4$
$y = - (x - 4)^2 - 2$
$y = - (x - 4) ^2 - 4$
Finally, they noticed that the left hand set of parabolas were a reflection of the right hand set in the vertical axis. Therefore, they reasoned that the parabola that went through $(- 4, 0)$ would be $y = - (x + 4)^2$ : the negative sign in front of the brackets produces the inverted parabola, and the $+ 4$ inside the bracket translates it $4$ units to the left.
As before, they noticed that $y = - (x + 4)^2$ could be raised or lowered to produce the rest of the parabolas in the left hand set by either adding or subtracting a $2$ or a $4$.
Therefore the solutions for the left hand set are:
$y = - (x + 4)^2$
$y = - (x + 4)^2 + 2$
$y = - (x + 4)^2 + 4$
$y = - (x + 4)^2 - 2$
$y = - (x + 4)^2 - 4$
Teachers' Resources
Why do this 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 and start to make generalisations.
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 $y=-(x-4)^2$.
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.
- $y = (x -4)^2$ is the same as $y = x^2$ but moved right four spaces.
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 Support
Students could begin by investigating translation of straight lines and look at how their equations change.
If students have made observations like those above, you could encourage them to start to generalise by asking about other pairs of curves that are the same except that one is flipped upside down, or where one is the same as the other but moved two spaces up or to the right, or three spaces, or another number of spaces.
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.