Mathematics: AA HL
Practice transformations
Worked example 1
Describe the sequence of transformations that transforms the graph of the reciprocal function y=
to the graph of the rational function y=
1
x
3 x +7
.
2 x−2
3 x +7
a
needs to be rewritten in the form
+c, because that form allows us to see
2 x−2
x−b
a
1
that the graph of y=
+c can be obtained by transforming the graph of y= in 3 steps:
x−b
x
The expression
1. Horizontal translation of b units to the right.
2. Vertical stretch with a factor a.
3. Vertical translation upward with a factor c.
In the numerator of y=
y=
3 x +7
, we need to force the denominator to appear:
2 x−2
3 x +7 a(2 x−2)+b
=
.
2 x−2
2 x−2
So, let 3 x +7 be a(2 x−2)+b
Expanding brackets:
3 x +7=2 ax−2 a+b
Comparing coefficients:
3=2 a
⇒ a=
7=−2 a+b
3
2
⇒
3
b=7+2⋅ =10
2
Therefore:
3
(2 x−2)+10
3 x +7
2
3
10
3
5
=
= +
= +
2 x−2
2 x−2
2 2 x−2
2 x−1
Now it is clear that the graph of y=
3 x +7
1
is derived from the graph of y= :
2 x−2
x
1. Horizontal translation of 1 unit to the right.
2. Vertical stretch with a factor 5.
3
3. Vertical translation upward with a factor .
2
These steps are graphed in Desmos: https://www.desmos.com/calculator/14dqjv7otb
Question 1
Describe the sequence of transformations that transforms the graph of the reciprocal function y=
to the graph of the rational function y=
1
x
x +7
.
3−5 x
The answer can be found in Desmos: https://www.desmos.com/calculator/cpnvrlzztv
Worked example 2
Describe the sequence of transformations that transforms the graph of y=ln ( x)
to the graph of y=ln (0.5 x +3)+2.
The equation can be rewritten as y=ln (0.5( x +6))+2 to separate the horizontal stretch from the
horizontal translation. This is the sequence of transformations:
1. Horizontal stretch with a factor 2. (The stretch factor is the reciprocal of 0.5.)
2. Horizontal translation 6 units to the left.
3. Vertical translation 2 units upward.
Please note that y=ln (0.5( x +6))+2 can be rewritten as y=ln ( x +6)+ln (0.5)+2. Therefore, this is
an alternative sequence of transformations:
1. Horizontal translation 6 units to the left.
2. Vertical translation 2+ln (0.5)=2−ln (2)≈1.307 units upward.
These steps are graphed in Desmos: https://www.desmos.com/calculator/jznqgcvbv0
Question 2
Describe the sequence of transformations that transforms the graph of y=ln ( x)
to the graph of y=ln ((2 x +3)3 )
The answer can be found in Desmos: https://www.desmos.com/calculator/rk7intuicu