SME
Grade 11 Math · Problem type
Graph of tangent
Use this for “Find the period and asymptotes of y = 2 tan(x/2).”
Chart
Vertical asymptotes every 2π
Period 2π ≈ 6.2832
f(π/2) = 2
This is an example. Type your own values to start a new problem.
y
=
a
tan(
b
x)
P
Period
worked out (not typed)
V
First asymptote right of 0
worked out (not typed)
x
Input (radians)
example
y
Output
worked out (not typed)
Formulas
P
=
π
b
2π
=
π
0.5
V
=
π
2 ×
b
π
=
π
2 × 0.5
y
=
a
×
tan(
b
×
x
)
2
=
2
×
tan(0.5
×
π
2
)
Assumptions
•
tan x repeats every π, so y = a tan(bx) has period P = π ÷ b.
•
It has no value where cos(bx) = 0: asymptotes at x = π ÷ (2b), then every period.
•
x is in radians.
Step-by-step
WE KNOW
a = 2, b = 0.5, x = π/2
FIND
period (P), first asymptote right of 0 (V), output (y)
Step 1 · Find period (P)
Use P = π ÷ b
Tangent
repeats
every
π;
b
fits
b
periods
into
each
π.
P
=
π
0.5
P
=
2π
(≈
6.28)
Step 2 · Find first asymptote right of 0 (V)
Use V = π ÷ (2 × b)
cos(bx)
=
0
first
at
bx
=
π
2
.
Then
every
period:
x
=
π
+
2πn
for
every
integer
n.
V
=
π
2 × 0.5
V
=
π
(≈
3.14)
Step 3 · Find output (y)
Use y = a × tan(b × x)
Multiply
x
by
b
,
take
the
tangent,
then
stretch
by
a
.
y
=
2
×
tan(0.5
×
π
2
)
y
=
2
×
tan(
π
4
)
y
=
2
×
1
y
=
2
Check
Put the numbers back in. Both sides should match.
2π
=
π
0.5
✓
π
=
π
2 × 0.5
✓
2
=
2
×
tan(0.5
×
π
2
)
✓
Topic check
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