TRANSFORMATIONS OF GRAPHS
1. The Modulus Function
, the modulus or absolute value of x, is the magnitude of x ignoring the sign. So we have
The graph of is identical to that ofwhen it is positive, but negative parts of the graph are reflected in the x-axis to become positive.
On a graphic calculator, use the Abs function (Optn → NUM →Abs) to generate the modulus. In Autograph, type or use the modulus button from the “add equation” window.
Example 1 :Sketch the graph of , where
Hence sketch the graphs of and .
The graph of is generated by taking the graph for positive x and reflecting in the y-axis.
This is because, for example , and so for all negative values of x, the graph behaves as it would for all the corresponding positive values of x.
Example2:Sketch the graph of , where
Hence sketch the graphs of and .
The graph of intersects the y-axis at (0, −9), and the x-axis at (3, 0), (−1, 0) and (−3, 0).
Example 3 : Sketch the graph of .
First we sketch the graph of , where . However we need only sketch the portion of the graph for which , as we can then reflect this in the y-axis.
Since, the graph crosses the x-axis at (0, 0) and (2, 0), and is an upside-down U-shape.
C3 p57 Ex 5A, p59 Ex 5B
2. Solving Equations Involving Modulus
Example 1 : Solve the equation .
At Q,
At P, the modulus sign has reflected to make . Therefore we have
The solution to the equation is
Example 2: Solve the equation .
Once again, we sketch and compare two graphs, namely and.
At P, the modulus has ‘reversed’ both of the graphs. So we have the equation
At Q, the modulus has only reversed the graph of . So we have the equation
So the solution to the equation is.
Example3:Solve the equation .
At P,
At Q,
So the solution to the equation is.
Example4:Solve the equation .
At P,
We ignore the negative root which is not appropriate here.
At Q,
Once again, we ignore the other root. So the solution to the equation is
NB : a straight line can be drawn which cuts the graph of in four places. In this case, all the roots of the quadratic equations would be relevant. As a challenge, find the range of values of k for which the line does this.
Example5:Let . Solve the equation .
Now . We draw the part of the function to the right of the y-axis, and reflect it across.
Next, we solve the equation .
The solution –2 is not appropriate here, as we can only take the function of a positive number.
So the solution to the equation is , corresponding to the points P and Q in the diagram.
C3 p62 Ex 5C
3. Summary of C1 Topic
In unit C1 we arrived at the following results…
transformation / maps on to...translation /
translation /
vertical stretch scale factor a /
horizontal stretch scale factor a /
reflection in the x-axis /
reflection in the y-axis /
180° rotation /
4. Combined Transformations
We now look at the effect of performing more than one transformation.
Example1:What transformations have to be performed on the graph of to obtain the graph of ?
transformation / graph has become...initially /
translation of /
vertical stretch, scale factor 3, x-axis invariant /
translation of /
Example2:What transformations have to be performed on the graph of to obtain the graph of?
Completing the square,
So the transformations are a translation of followed by a vertical stretch, scale factor 2, with the x-axis invariant.
Note that we could have written the quadratic as , which would have given us a translation of followed by a vertical stretch, scale factor 2, with the x-axis invariant, followed by a translation of .
Example3:Find the equation of the graph obtained when the graph of is first reflected in the y-axis, and then translated by +2 units in the direction of the x-axis.
transformation / graph has become...initially /
reflection in the y-axis /
translation of /
Example 4 : Sketch the graph of .
Example 5 : Sketch the graph of .
Example 6 : Sketch the graph of
Example7:The function has period 4 and is defined by
Taking the x-axis from –4 to 4, sketch the graphs of
a)
b)
c)
a)
b) We translate the graph one unit to the left.
c)The equation rearranges to So we reflect our original graph in the x-axis, stretch it vertically scale factor , and horizontally scale factor 2.
Example8:The function is an odd function (this means it has 180° rotational symmetry about the origin) and is defined by
Taking the x-axis from –4 to 4, sketch the graphs of
a)
b)
c)
a)
b) We translate the graph one unit to the right and two units down.
c) We reflect the graph in the y-axis.
Example 9 : The diagram shows a sketch of the graph .
Sketch the graphs of
a) b) c) d)
in each case finding the coordinates of the images of A, B and O.
a) We perform a vertical stretch scale factor 2, followed by a translation 1 unit down.
b) We perform a translation 2 units left and one unit up.
c) We perform a horizontal stretch and a vertical stretch, both scale factor .
d) We perform a translation 1 unit right, and then reflect in the x-axis.
C3 p67 Ex 5D,p69 Ex 5E Topic Review : Transformations of Graphs