1. Appearance of Different Curves
Slideshow 26, Mathematics
Mr. Richard Sasaki

2. Objectives Review graphs that we have learned so far
Understand the appearance of different curves that touch the origin
Understand the difference in appearance for graphs in the form 𝑦=𝑎 𝑥 2 where 𝑎>0 and 𝑎<0

3. So far… We know the appearance of three graphs… 𝑦= 𝑥 2 2 𝑦= 𝑥 2
𝑦= 𝑥 2 2
𝑦= 𝑥 2
𝑦= 2𝑥 2

4. increases
lower
𝑦= 𝑥 2 3
𝑦= 𝑥 2 2
𝑦=8 𝑥 2
𝑦=17 𝑥 2
𝑦= 𝑥 2
𝑦=3 𝑥 2
The line is completely horizontal (linear) and touching the 𝑥 – axis at all points.
The line will curve in the opposite direction and remain in quadrants III and IV.

5. 𝑦=𝑎 𝑥 2 As 𝑎 increases and decreases, we see this effect…
Note: The moving graph shown increments in smaller values of 𝑎 when 𝑎 is close to 0.

6. 𝑦=𝑎 𝑥 2
The curve that 𝑦=𝑎 𝑥 2 (and other quadratics in the form 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐, and other things too…) forms is called a
parabola
The parabola is always
symmetrical
The point at the bottom of a parabola is called the
minimum
It is also known as the
vertex

7. 𝑦=𝑎 𝑥 2 The plural of parabola is or . parabolae parabolas
Parabolae can also curve in the other direction.
The point at the top of a parabola is called the
maximum
It is also known as the
vertex

8. Drawing Parabolae
Drawing a parabola neatly is important. It’s not so difficult when it’s in the form 𝑦=𝑎 𝑥 2 but may be troublesome for lazy students.
Plotting at least 3 points minimum is necessary. The maximum / minimum and two points either side of this. Which points you choose is up to you.
The problem is, if you just plot three points, the graph can still have different appearances. We should know how parabolae look and apply that.

9. Drawing Parabolae So if we have these three points for 𝑦=2 𝑥 2 …
Is this good?
This is better. The V shape is fine but the rate of change doesn’t grow so steeply.

10. Answers
𝑦= 𝑥 2
𝑦= 1 2 𝑥 2

11. Answers
𝑦= −2𝑥 2
𝑦= 8𝑥 2

12. Answers Why do the minima / maxima always touch the origin?
For any relationship in the form 𝑦=𝑎 𝑥 2 , when 𝑥=0, 𝑦=0, so the line drawn for the relationship would pass through (0, 0).
Give an example of a non-linear graph that does not go through the origin.
Anything in the form 𝑦=𝑎 𝑥 2 +𝑏 would be fine where 𝑏≠0, ask me to check other answers!