1. An Introduction to Graphing Squared Variables
Slideshow 25, Mathematics
Mr. Richard Sasaki

2. Objectives
Review Direct Proportion and understand the meaning of a variable
Plot the graph 𝑦= 𝑥 2
Understand the meaning of the way the line is formed for 𝑦= 𝑥 2

3. Direct Proportion What is direct proportion?
Direct proportion is a relationship between two variables where the relationship is a coefficient of one of the variables.
If 𝑦∝𝑥, we can say that
𝑦=𝑘𝑥
If 𝑦∝ 𝑥 𝑎 , we can say that
𝑦=𝑘 𝑥 𝑎

4. Direct Proportion
So, as you know, in direct proportion, we usually use 𝑥 and 𝑦 which are variables. What are variables?
A variable is represented by an unknown which can take different values. The value is not constant.
If 𝑦∝𝑥, 𝑦=𝑘𝑥 where….
𝑥 and 𝑦 are and 𝑘 is a
variables
constant
The same applies for 𝑦∝ 𝑥 𝑎 where 𝑎∈ ℤ + .

5. Plotting 𝑦= 𝑥 2
I am sure a lot of you know what 𝑦= 𝑥 2 looks like so we will go through this quickly.
To plot it, we need to make a table considering certain likely values. Let’s consider values −5≤𝑥≤5, 𝑥∈ℤ.
𝑦= 𝑥 2 𝒙 −5 −4 −3 −2 −1 1 2 3 4 5 𝒚 𝟐𝟓 𝟏𝟔 𝟗 𝟒 𝟏 𝟎 𝟏 𝟒 𝟗 𝟏𝟔 𝟐𝟓
A piece of paper will come around now. Plot these points, don’t join them!

6. Plotting 𝑦= 𝑥 2 x −1
−0.75
−0.5
−0.25
0.25
0.5
0.75 1 y
0.0625
0.5625
0.25
0.0625
0.25
0.5625
Now, if you did connect the dots with straight lines, it’d look like this:
Fantastic, it’s brilliant…
or not.
To see how it actually looks, we need to be more accurate, especially around 0, 0 .
Please copy these points and plot them!

7. Plotting 𝑦= 𝑥 2
Hopefully, you should be able to see that they form a curve…
Try your best to join the dots on your graph. (Do not use straight line segments.)
I hope yours is better than mine! Answer the questions about the curve at the bottom of the worksheet!

8. The Bottom Part
As the line drawn above is not straight, the relationship between 𝑦 and 𝑥 is not
linear
The graph is only present in quadrants I and II because 𝑦 is always or equal to zero.
greater than
The graph is about the 𝑦 axis.
symmetrical
For 𝑥>0, as 𝑥 increases, 𝑦
increases
For 𝑥<0, as 𝑥 increases, 𝑦
decreases
The minimum is at the graph’s
origin

9. Plotting 𝑦= 2𝑥 2
When we plot a graph in the form 𝑦=𝑎 𝑥 2 , how does 𝑎 influence its appearance?
Next, let’s try 𝑦=2 𝑥 2 and see how it looks different.
First, we need to consider the table of values. Extra values should be considered close to 𝑥=0. 𝒙 −3 −2 −1
−0.5
−0.25
0.25
0.5 1 2 3 𝒚 18 8 2
0.5
0.125
0.125
0.5 2 8 18

10. Plotting 𝑦= 2𝑥 2 𝒙 −3 −2 −1
−0.5
−0.25
0.25
0.5 1 2 3 𝒚 18 8 2
0.5
0.125
0.125
0.5 2 8 18
First, plot these points and draw the graph.
Try your best to make it curve like the graph shown here.
Next answer the questions below!

11. The Bottom Part
The graph 𝑦=2 𝑥 2 only exists in quadrants and because 𝑦 is never I II
negative
At all points, the line 𝑦= 𝑥 2 is than the line 𝑦=2 𝑥 2 except for where they meet at the
wider
origin
For equal values of 𝑥, the values of 𝑦 where 𝑦=2 𝑥 2 are that of the values of 𝑦 where 𝑦= 𝑥 2 at all points.
double
For 𝑦= 1 2 𝑥 2 , the rate of change at all points is than 𝑦= 𝑥 2 .
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