# Fst 4.9 Graph – Standardization Theorem ```Section 4.9

In a relation described by a sentence in x and
y, the following processes yield the same
graph:
xh
yk
 1. replacing x by
and y by
in the
a
b
sentence;

2. Applying the scale change (x,y)  (ax,by)
where a, b≠ 0, followed by the applying the
translation (x,y)  (x + h, y + k) to the graph
of the original relation.





yk
xh
 sin(
) has
b
a
Notice : to use this formula,
Amp: |b|
Period: 2π *|a| x must be by itself!
Phase shift: h
Vertical shift: k

Describe the graph of y = sin (2x – π/4)
y = sin (2(x – π/8))
Amp: 1
Period: 2π/2 = π
Phase shift: π/8 to the right
Vertical shift: 0

The composite of what scale change and
translation maps y = cos x onto
y = 3 cos 4(x + π) – 10
S(x,y)  (x/4, 3y)
T(x,y)  (x – π, y – 10)
x4
y  2 sin(
)
4
S(x,y)  (4x, 2y)
T(x,y)  (x + 4, y)
Amp  2
Period  2π * 4 = 8π
Vertical shift  0
Phase shift  4 to the right
y = 2 sin (3x + π) + 2
y = 2 sin 3(x + π/3) + 2
S(x,y)  (x/3, 2y)
T(x,y)  (x - π, y + 2)
Amp  2
Period  2π * 1/3 = 2π/3
Vertical shift  2 up
Phase shift π/3 to the left
Pages 290
1 , 3 – 6,
9 – 11
*Don’t need to graph
```