SME
Grade 11 Math · Algebra 2
Graphs of sine, cosine and tangent
Use this for “Find the amplitude, period, phase shift and midline of y = 3 sin(2(x − π/4)) + 1.”
Chart
Local maximum at (π/2 ≈ 1.5708, 4)
Local minimum at (π ≈ 3.1416, −2)
Midline y = 1
Amplitude 3
Period π ≈ 3.1416
f(π/2) = 4
This is an example. Type your own values to start a new problem.
y
=
a
sin(
b
(x
−
h
))
+
k
A
Amplitude
worked out (not typed)
P
Period
worked out (not typed)
x
Input (radians)
example
y
Output
worked out (not typed)
Formulas
A
=
|
a
|
3 = |3|
P
=
2π
b
π
=
2π
2
y
=
a
×
sin(
b
×
(
x
−
h
))
+
k
4
=
3
×
sin(2
×
(
π
2
−
π
4
))
+
1
Assumptions
•
The midline is y = k; the graph rises and falls A = |a| above and below it.
•
b fits b cycles into every 2π, so the period is P = 2π ÷ b.
•
h slides the start of the cycle to x = h; x is in radians.
Step-by-step
WE KNOW
a = 3, b = 2, h = π/4, k = 1, x = π/2
FIND
amplitude (A), period (P), output (y)
Step 1 · Find amplitude (A)
Use A = |a|
The amplitude is a distance, so it is never negative.
A
=
|3|
A
=
3
Step 2 · Find period (P)
Use P = 2π ÷ b
One
full
turn,
2π,
shared
among
b
cycles.
P
=
2π
2
P
=
π
(≈
3.14)
Step 3 · Find output (y)
Use y = a × sin(b × (x − h)) + k
Shift
x
by
h
,
multiply
by
b
,
take
the
sine,
stretch
by
a
,
then
add
k
.
y
=
3
×
sin(2
×
(
π
2
−
π
4
))
+
1
y
=
3
×
sin(
π
2
)
+
1
y
=
3
×
1
+
1
y
=
4
Check
Put the numbers back in. Both sides should match.
3 = |3| ✓
π
=
2π
2
✓
4
=
3
×
sin(2
×
(
π
2
−
π
4
))
+
1
✓
Topic check
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