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Unit 4 · Trigonometric Functions

4.6b Graphs of Sine &
Cosine (Part 2)

Objective: Graph sine and cosine functions using the 5 key points, amplitude, period, phase shift, and vertical translation.

5 Key Points Amplitude Period Phase Shift Vertical Translation
Concept

5 Key Points from Parent Graphs

The key to graphing any sine or cosine curve is to be familiar with the 5 key points from their parent graphs.

\(y = \sin(x)\)
1 -1 0 π/2 π 3π/2 2π middle high middle low middle (0, 0) (π/2, 1) (π, 0) (3π/2, -1) (2π, 0)
\(x\)\(0\)\(\frac{\pi}{2}\)\(\pi\)\(\frac{3\pi}{2}\)\(2\pi\)
\(\sin x\)\(0\)\(1\)\(0\)\(-1\)\(0\)
\(y = \cos(x)\)
1 -1 0 π/2 π 3π/2 2π high middle low middle high (0, 1) (π/2, 0) (π, -1) (3π/2, 0) (2π, 1)
\(x\)\(0\)\(\frac{\pi}{2}\)\(\pi\)\(\frac{3\pi}{2}\)\(2\pi\)
\(\cos x\)\(1\)\(0\)\(-1\)\(0\)\(1\)
Reference

Steps for Graphing a Sine or Cosine Function

1
T-TABLE. Write down a t-table with 5 points for the parent graph.
2
ANALYZE. Determine the transformations of the function.
Affects the \(x\)-coordinates
period • phase shift
Affects the \(y\)-coordinates
amplitude • vertical translation
3
NEW X-COORDINATES. Use the period, then phase shift to adjust the x-coordinates from the parent graph's t-table.
4
NEW Y-COORDINATES. Use the amplitude, then vertical translation to adjust the y-coordinates from the parent graph's t-table.
5
Set up axes, scale, plot points, and draw the graph!
Important
The order matters: multiply before add/subtract!
Example 1

Graphing a Sine Function

Graph \(y = 3\sin\!\left(\dfrac{x}{2}\right)\).
Amplitude
\[|a| = 3\] Multiply \(y\)'s by 3
Period
\[\frac{2\pi}{b} = \frac{2\pi}{1/2} = 4\pi\] Multiply \(x\)'s by 2 because \(b = \tfrac{1}{2}\)
Phase Shift & Vertical Translation
Phase shift: none     Vertical translation: none
T-Tables
Parent sin
\(x\)\(0\)\(\frac{\pi}{2}\)\(\pi\)\(\frac{3\pi}{2}\)\(2\pi\)
\(\sin x\)\(0\)\(1\)\(0\)\(-1\)\(0\)
→
Transformed
\(x\)\(0\)\(\pi\)\(2\pi\)\(3\pi\)\(4\pi\)
\(y\)\(0\)\(3\)\(0\)\(-3\)\(0\)
\(x\)'s × 2  |  \(y\)'s × 3
Graph
3 -3 π 2π 3π 4π 5π 6π 7π 8π one period Repeat the pattern for another period
Summary
\[y = 3\sin\!\left(\tfrac{x}{2}\right)\quad\text{Amplitude } 3,\;\text{Period } 4\pi\]
Example 2

Graphing with Phase Shift & Reflection

Graph \(y = -\sin\!\left(x + \dfrac{\pi}{2}\right) + 1\).
Amplitude
\[|a| = 1,\;\text{reflection in }x\text{-axis}\] Multiply \(y\)'s by \(-1\)
Period
\[\frac{2\pi}{b} = \frac{2\pi}{1} = 2\pi\] No horizontal stretch/shrink
Phase Shift
\[\frac{\pi}{2}\text{ to the left}\] Subtract \(\frac{\pi}{2}\) from \(x\)'s
Vertical Translation
\[\text{Up } 1\] Add 1 to \(y\)'s
T-Tables
Parent sin
\(x\)\(0\)\(\frac{\pi}{2}\)\(\pi\)\(\frac{3\pi}{2}\)\(2\pi\)
\(\sin x\)\(0\)\(1\)\(0\)\(-1\)\(0\)
→
Transformed
\(x\)\(-\frac{\pi}{2}\)\(0\)\(\frac{\pi}{2}\)\(\pi\)\(\frac{3\pi}{2}\)
\(y\)\(1\)\(0\)\(1\)\(2\)\(1\)
\(y\)'s × (\(-1\)), then + 1  |  \(x\)'s \(-\) \(\frac{\pi}{2}\)
Graph
y=1 0 1 2 -1 -π/2 π/2 π 3π/2 2π 5π/2 3π 7π/2
Summary
\[y = -\sin\!\left(x + \tfrac{\pi}{2}\right) + 1\quad\text{Amp } 1,\;\text{Period } 2\pi,\;\text{Shift } \tfrac{\pi}{2}\text{ left},\;\text{Up } 1\]
Example 3

Graphing with All Transformations

Graph \(y = \dfrac{1}{2}\cos(2x - 2\pi) - 1\).   Rewrite: \(y = \dfrac{1}{2}\cos\!\bigl(2(x - \pi)\bigr) - 1\)
Amplitude
\[|a| = \tfrac{1}{2}\] Multiply \(y\)'s by \(\frac{1}{2}\)
Period
\[\frac{2\pi}{b} = \frac{2\pi}{2} = \pi\] Multiply \(x\)'s by \(\frac{1}{2}\) because \(b = 2\)
Phase Shift
\[\pi\text{ to the right}\] Add \(\pi\) to \(x\)'s
Vertical Translation
\[\text{Down } 1\] Subtract 1 from \(y\)'s
T-Tables
Parent cos
\(x\)\(0\)\(\frac{\pi}{2}\)\(\pi\)\(\frac{3\pi}{2}\)\(2\pi\)
\(\cos x\)\(1\)\(0\)\(-1\)\(0\)\(1\)
→
Transformed
\(x\)\(\pi\)\(\frac{5\pi}{4}\)\(\frac{3\pi}{2}\)\(\frac{7\pi}{4}\)\(2\pi\)
\(y\)\(-\frac{1}{2}\)\(-1\)\(-\frac{3}{2}\)\(-1\)\(-\frac{1}{2}\)
\(x\)'s × \(\frac{1}{2}\), then + \(\pi\)  |  \(y\)'s × \(\frac{1}{2}\), then \(-\) 1
Graph
y=-1 1 0 -1/2 -1 -3/2 π 5π/4 3π/2 7π/4 2π 9π/4 5π/2 11π/4 3π Period 1 Period 2
Summary
\[y = \tfrac{1}{2}\cos\!\bigl(2(x-\pi)\bigr) - 1\quad\text{Amp } \tfrac{1}{2},\;\text{Period } \pi,\;\text{Shift } \pi\text{ right},\;\text{Down } 1\]