Drawing Quadratic Curves Slideshow 27, Mathematics Mr. Richard Sasaki

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

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

𝑦=𝑎 𝑥 2 +𝑏 If we add a constant 𝑏 to the statement, what effect does it have? 𝑦= 𝑥 2 2 +1 Example Draw the graph 𝑦= 𝑥 2 2 +1. 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

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

Answers – Easy (Bottom) 𝑦= 𝑥 2 2 +1 𝑦= −𝑥 2 +5 For a graph in the form 𝑦=𝑎 𝑥 2 +𝑏, it’s vertex is at ( , ). 0 𝑏

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

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

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

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

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−2 2 +1 𝑦= 𝑥−6 2 +4

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

Answers – Medium (Top) Write down the vertex as a pair of co-ordinates for 𝑦=2 𝑥+1 2 −3. (−1, −3) 𝑦= − 𝑥−2 2 +1 𝑦= 2 𝑥+3 2 +4

Answers – Medium (Bottom) 𝑦= 𝑥−2 2 2 −3 𝑦= 𝑥+1 2 3 −4

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.