Unit 4
MCR 3U1
Lesson 5: Transformations of Exponential Functions – Part 1
Remember the general equation f ( x)   af ( k ( x  d ))  c ? It’s back!!!
The general form of an exponential equation is:
f ( x)  ab k ( x  d )  c , where each letter in
the general equation represents a transformation.
Stretches
If a  1 , a vertical expansion by a factor of a occurs.
If 0  a  1, a vertical compression by a factor of a occurs.
Example 1:
y  4( 2 x )
1
y  (2 x )
2
__________ by a factor of ____
__________ by a factor of ____
1
If
k  1, then a horizontal compression by a factor of k
If
k  1, then a horizontal expansion by a factor of
occurs.
1
occurs.
k
Example 2:
y  22 x
____________ by a factor of _____
1
x
22
____________ by a factor of _____
y
Reflections
If a  0 , then the function is reflected in the x-axis.
If
k  0 , then the function is reflected in the y-axis.
Example 3:
y  2(3 x )
Reflection in the _____-axis
y  3 4 x
Reflection in the _____-axis
Translations
If
d  0 , a horizontal translation of d units to the left occurs.
d  0 , a horizontal translation of d units to the right occurs.
If
If
c  0 , a vertical translation of c units upwards occurs.
c  0 , a vertical translation of c units downwards occurs.
If
Unit 4
MCR 3U1
Example 3:
y  2 x 1
Translated to the _________
y  2 x 1
Translated to the _________
y  2x 1
Translated ______________
y  2x  1
Translated ______________
When performing transformations, do them in the same order as with quadratic functions:
1. Compressions and Expansions
2. Reflections
3. Translations
You may have to factor the exponent to see what the transformations are. For example, if the
exponent is 3x + 3, it is easier to see that there was a horizontal stretch of 3 and a horizontal
translation to the left 1 unit if you factor the expression first.
The domain is always x  R for exponential functions. The range depends on the location of
the horizontal asymptote. If it is above the asymptote, its range is y  c . If it is below the
asymptote, its range is
y  c.
The equation of the asymptote is
y  c.
Example 4: Describe the properties of the equation y  4 x  3 .
Example 5: Use transformations to sketch the graph of y  32 x  2  4 .