1. Drawing Quadratic Curves
Slideshow 27, Mathematics
Mr. Richard Sasaki

2. Objectives Understand how to draw graphs in the form 𝑦=𝑎 𝑥 2 +𝑏
Learn how to draw graphs in the form 𝑦=𝑎 𝑥−ℎ 2 +𝑘 where 𝑎≠0

3. Review We know that for a graph in the form 𝑦=𝑎 𝑥 2 … For 𝑎>0…
When 𝑎 is small it looks like…
When 𝑎 is large it looks like…
When 𝑎 is small it looks like…
When 𝑎 is large it looks like…
𝑥 2 4
− 𝑥 2 3 𝑦= 𝑦=
5 𝑥 2 𝑦= 𝑦=
−6 𝑥 2

4. 𝑦=𝑎 𝑥 2 +𝑏
If we add a constant 𝑏 to the statement, what effect does it have?
𝑦= 𝑥
Example
Draw the graph 𝑦= 𝑥
We know its vertex is at ( , ) and the line is within Quadrants and .
0 1
𝐼 𝐼𝐼 𝒙 −𝟐 𝟎 𝟐 𝑦 1 3 3

5. Answers – Easy (Top)
𝑦= 𝑥 2 −1
𝑦= 2𝑥 2 +2

6. Answers – Easy (Bottom)
𝑦= 𝑥
𝑦= −𝑥 2 +5
For a graph in the form 𝑦=𝑎 𝑥 2 +𝑏, it’s vertex is at ( , ).
0 𝑏

7. Answers – Hard (Top)
𝑦= 2𝑥 2 − 1 2
𝑦= 4𝑥 2 −3

8. Answers – Hard (Bottom)
𝑦= 𝑥
𝑦= −3𝑥 2 −2
If both axes range from −100 to 100, the rate of change will appear greater (line looks steeper).

9. In the form 𝑦=𝑎 𝑥−ℎ 2 +𝑘
For a graph with the equation 𝑦=𝑎 𝑥−ℎ 2 +𝑘, what are the co-ordinates of its vertex?
In this form, and are constants. This makes the curve positive if is positive and negative if is negative. ℎ 𝑘 𝑎 𝑎
The vertex must be a minimum if and a maximum if
𝑎>0
𝑎<0
What is the smallest value 𝑦 can be if 𝑎>0? 𝑘
What is the highest value 𝑦 can be if 𝑎<0? 𝑘
Both of the above occur when = .
𝑥 ℎ
So the co-ordinates when 𝑥=ℎ and 𝑦=𝑘 are ( , ).
ℎ 𝑘

10. 𝑦=𝑎 𝑥−ℎ 2 +𝑘 (Vertex Form)
Example
𝑦=2 𝑥−3 2 −1
Draw the graph 𝑦=2 𝑥−3 2 −1 and state its vertex.
We know its vertex is at ( , ) and its shape is positive.
3 −1 𝒙 𝟐 𝟑 𝟒 𝑦 −1 1 1

11. Answers – Easy (Top) 𝑦= 𝑥−2 2 +1 𝑦= 𝑥−6 2 +4
A graph in the form 𝑦=𝑎 𝑥−ℎ 2 +𝑘 has a vertex at point ( , ) .
ℎ 𝑘
𝑦= 𝑥−
𝑦= 𝑥−

12. Answers – Easy (Bottom)
𝑦= 2 𝑥−3 2 −1
𝑦= 3 𝑥−

13. Answers – Medium (Top)
Write down the vertex as a pair of co-ordinates for 𝑦=2 𝑥+1 2 −3.
(−1, −3)
𝑦= − 𝑥−
𝑦= 2 𝑥

14. Answers – Medium (Bottom)
𝑦= 𝑥− −3
𝑦= 𝑥 −4

15. Answers – Hard
1. 2 𝑥+2 2 −1, (−2, −1)
2. A higher value of ℎ shifts the graph to the right and a lower value shifts it to the left.
3. The axis of symmetry exists where 𝑥=ℎ.