1.6 Exploring Transformations of Parent Functions
Investigation:
The function defined by g(x) = af(x-d)+c describes the transformations of the graph
of f(x).
When f(x) =x2,
g(x) = a(x-d)2 + c
When f(x)= √𝑥
When f(x) =
1
,
g(x) =
,
g(x) =
When f(x) = |x|,
g(x) =
𝑥
Exploring vertical stretches, compressions and reflections:
Example 1.
Copy and complete tables of values for
a) f(x) = x2
c) f(x) = 1/2x2
b) f(x) = 3x2
d) f(x) = -2x2
Describe the transformations in words:
a) No change- Parent Function
b) Vertical stretch by a factor of 3
c) Vertical compression by a factor of 0.5
d) Reflection in the x- axis; vertical stretch by a factor of 2.
Example 2
Graph the following functions on the same set of axes:
a) f(x) =√x
b) f(x)= 3√x
c) f(x)=1/2√x
d) f(x)=-2√x
Describe the transformations in words:
a) No change- Parent Function
b) Vertical stretch by a factor of 3
c) Vertical compression by a factor of 0.5
d) Reflection in the x- axis; vertical stretch by a factor of 2.
Example 2
Sketch f(x)= 3x2 + 2 and f(x) = 3x2 - 1 on the same set of axes. Describe the
transformations for each transformed function in words.
Example 4: Given g(x) = |x|,
a) write equations for :
i) g(x - 2)
ii) 0.5g(x + 1)
iii) 3g(x - 1) + 2
g(x) = |x – 2|
g(x) = 3|x – 1| + 2
g(x) = 0.5|x + 1|
b) Complete the tables of values for each of these transformations (start with the parent
function g(x)= |x| first!)
c) Graph the transformed functions on the same set of axes:
Reflecting:
a) When you graphed y = af(x - d) + c, what were the effects of c and d?
b) How did the graphs with a> 1 compare with the graphs with a<1?
c) How did the graphs with a > 1 compare with those with 0< a <1?