Transformations
Transformations are the key to graphing and explaining where the parabola is. It is only used in vertex form because each letter except x and y represents a transformation in this equation y=a(x-h)^2+k.
h = the vertex of the parabola will move to the right or left side of the graph. (Negative numbers move right and positive numbers move left) This is called a horizontal translation right or left depending on the way it goes.
k = the vertex of the parabola will move up or down. (Positive numbers move up and negative numbers move down) This is called a vertical translation up or down depending on the way it goes.
a = how steep the graph is. If a is higher than one the graph will get steeper. This is called a vertical stretch.
If a is less than one but higher than zero the graph will become less steep. This is called a vertical compression.
If a is a negative the parabola will flip. This is called a vertical reflection because the parabola is the same but is just flipped, therefore making it a reflection.
Therefore there are five transformations:
The first one is a horizontal translation.
The second one is a vertical translation.
The third one is a vertical compression.
The forth one is a vertical stretch.
The fifth one is a vertical reflection.
Here is an example of a transformation question:
Describe, using appropriate terminology, the transformations that occurred to y=x^2 in order to become y=-2(x-3)^2+5
First there is a vertical reflection because of the - in front of the 2.
Second there is a horizontal translation right 3 because of the -3.
Third there is a vertical translation up 5 because of the +5.
Last there is a vertical stretch of 2 because of the 2 in front of the bracket.
h = the vertex of the parabola will move to the right or left side of the graph. (Negative numbers move right and positive numbers move left) This is called a horizontal translation right or left depending on the way it goes.
k = the vertex of the parabola will move up or down. (Positive numbers move up and negative numbers move down) This is called a vertical translation up or down depending on the way it goes.
a = how steep the graph is. If a is higher than one the graph will get steeper. This is called a vertical stretch.
If a is less than one but higher than zero the graph will become less steep. This is called a vertical compression.
If a is a negative the parabola will flip. This is called a vertical reflection because the parabola is the same but is just flipped, therefore making it a reflection.
Therefore there are five transformations:
The first one is a horizontal translation.
The second one is a vertical translation.
The third one is a vertical compression.
The forth one is a vertical stretch.
The fifth one is a vertical reflection.
Here is an example of a transformation question:
Describe, using appropriate terminology, the transformations that occurred to y=x^2 in order to become y=-2(x-3)^2+5
First there is a vertical reflection because of the - in front of the 2.
Second there is a horizontal translation right 3 because of the -3.
Third there is a vertical translation up 5 because of the +5.
Last there is a vertical stretch of 2 because of the 2 in front of the bracket.