MHF 4U1 Transformations
SUMMARY – TRANSFORMATIONS
Compression and Stretching of Functions
Suppose that y = f(x) is a function, k > 1, a > 1.
 The graph of y = f(kx) is obtained by compressing horizontally the
graph of y = f(x) by a factor of k.
𝑥
 The graph of y = 𝑓 ( ) is obtained by stretching horizontally the graph
𝑘
of y = f(x) by a factor of k.
 The graph of y = af(x) is obtained by stretching vertically the graph of
y = f(x) by a factor of a.
𝑓(𝑥)
 The graph of y =
is obtained by compressing vertically the graph
𝑎
of y = f(x) by a factor of a.
Translations of Functions
Suppose that y = f(x) is a function, p > 0, q > 0
 The graph of y = f(x – p) is obtained by translating the graph of y = f(x)
p units to the right;
 The graph of y = f(x + p) is obtained by translating the graph of y =
f(x) p units to the left;
 The graph of y = f(x) + q is obtained by translating the graph of y =
f(x) q units upwards.
 The graph of y = f(x) – q is obtained by translating the graph of y = f(x)
q units downwards.
Reflections
Suppose that y = f(x) is a function; then
 The graph of y = f(–x) is obtained by reflecting the graph of y = f(x)
across the y-axis;
The graph of y = –f(x) is obtained by reflecting the graph of y = f(x)
across the x-axis.
Practice
1. Describe how the graph each transformed function compares to the
graph of the original (second function).
a. y = – (x – 4)2– 2
1
y = x2
b. y = √2(𝑥 + 1)
𝑦 = √𝑥
c. y = – √−2𝑥 + 6 − 3
𝑦 = √𝑥
d. y = 2 (x – 4)2– 1
y = (𝑥 + 1)2 – 3
2
2. Write the image equation given the following instructions:
a. The graph of 𝑦 = √𝑥 is stretched vertically by a factor of 3, reflected
in the x-axis, translated 3 units right and 2 units down.
b. The graph of y = |x| is stretched horizontally by a factor of 2, reflected
in the y-axis, translated 8 units left.
3. Use mapping notation to graph the following transformed functions.
State the domain and range of each image function:
a) y = – (x – 4)2
b) y = 2(𝑥 + 3)2 – 4
c) y = √−(𝑥 + 4) + 2
d) y = √−𝑥 + 5 − 3
e) y = 3 |x + 5| – 4
f) 𝑦 =
1
𝑥+ 2
+ 3