Graphing Sine Functions: Amplitude, Period, Phase Shift & Sketching

Graph of Sine, \( a \sin(bx + c) \), Function

Graphing and sketching sine functions of the form: \[ f(x) = a \sin(bx + c) \] A step-by-step tutorial is presented. Properties such as domain, range, period, phase shift, and intercepts of the graphs of these functions are examined in detail.

Once you finish the present tutorial, you may want to go through a self-test on trigonometric graphs. Free graph paper is available.

Review: Properties of the Basic Sine Function

We first start with the graph of the basic sine function:

\[ f(x) = \sin(x) \]

Domain: The set of all real numbers \( (-\infty, +\infty) \).
Range: The interval \( [-1, 1] \), expressed as: \[ -1 \le \sin(x) \le 1 \]
Period: Periodic with period equal to \( 2\pi \).

The graph of \( f \) over one period can be sketched by finding key points that provide crucial information such as \( x \)-intercepts, \( y \)-intercept, maxima, and minima.

\( x \)	\(0\)	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
\( f(x) = \sin(x) \)	\(0\)	\(1\)	\(0\)	\(-1\)	\(0\)

These 5 key points divide the period into 4 equal parts, making it easy to sketch the curve:

Key points of sine function

Graph of sin(x)

To understand why the graph of \( \sin(x) \) changes with \( x \) as shown above, you may want to review the interactive tutorial on the trigonometric unit circle.

Graphing \( f(x) = a \sin(bx + c) \)

We need to understand how the parameters \( a \), \( b \), and \( c \) affect the graph of \( f(x) = a \sin(bx + c) \) compared to \( \sin(x) \). You can also explore this interactively via sine functions applets.

Amplitude & Range: The range of \( bx + c \) is all real numbers, so the range of \( \sin(bx + c) \) is \( [-1, 1] \). Multiplying by \( a \): \[ -|a| \le a \sin(bx + c) \le |a| \] Thus, the range of \( f \) is \( [-|a|, |a|] \), and \( |a| \) is called the amplitude.
Period: For \( f(x) = a \sin(bx) \) to complete one cycle, the argument \( bx \) must vary from \( 0 \) to \( 2\pi \): \[ 0 \le bx \le 2\pi \implies 0 \le x \le \frac{2\pi}{|b|} \] Therefore, the period is given by: \[ \text{Period} = \frac{2\pi}{|b|} \]
Phase Shift: Considering the full argument \( bx + c \): \[ 0 \le bx + c \le 2\pi \implies -\frac{c}{b} \le x \le \frac{2\pi - c}{b} \] The constant \( c \) does not affect the period, but introduces a horizontal shift of \( -\dfrac{c}{b} \), which is called the phase shift.
- If \( -\dfrac{c}{b} \lt 0 \), the shift is to the left.
- If \( -\dfrac{c}{b} \gt 0 \), the shift is to the right.

Phase shift -c/b < 0 Phase shift -c/b > 0

Examples & Step-by-Step Solutions

Example 1: Sine Wave with Amplitude, Frequency & Phase Shift \( f(x) = 2\sin\left(3x - \dfrac{\pi}{2}\right) \)

Problem: Let \( f \) be a function given by \( f(x) = 2\sin\left(3x - \dfrac{\pi}{2}\right) \).

Find the domain and range of \( f \).
Find the period and the phase shift of the graph of \( f \).
Sketch the graph of function \( f \) over one period.

Solution:

Domain & Range: The domain is all real numbers \( (-\infty, +\infty) \). The range is given by the interval \( [-2, 2] \).
Period & Phase Shift:
* Period: \( \dfrac{2\pi}{|b|} = \dfrac{2\pi}{3} \).
* Phase shift: \( -\dfrac{c}{b} = -\left(-\dfrac{\pi}{2}\right) \Big/ 3 = \dfrac{\pi}{6} \) (shifted \( \dfrac{\pi}{6} \) units to the right).

Sketching via Key Points:
Let the argument \( 3x - \dfrac{\pi}{2} \) vary from \( 0 \) to \( 2\pi \) to cover one complete period:

\( 3x - \dfrac{\pi}{2} \)	\(0\)	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
\( f(x) \)	\(0\)	\(2\)	\(0\)	\(-2\)	\(0\)

Solving for \( x \) from \( 0 \le 3x - \dfrac{\pi}{2} \le 2\pi \): \[ \dfrac{\pi}{2} \le 3x \le \dfrac{5\pi}{2} \implies \dfrac{\pi}{6} \le x \le \dfrac{5\pi}{6} \] Unreduced fractional step intervals (\( \dfrac{\pi}{6} \) increments) make scaling the \( x \)-axis straightforward:

\( 3x - \dfrac{\pi}{2} \)	\(0\)	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
\( f(x) \)	\(0\)	\(2\)	\(0\)	\(-2\)	\(0\)
\( x \)	\(\dfrac{\pi}{6}\)	\(\dfrac{2\pi}{6}\)	\(\dfrac{3\pi}{6}\)	\(\dfrac{4\pi}{6}\)	\(\dfrac{5\pi}{6}\)

Plotting the points \( (x, f(x)) \) and joining them with a smooth sine wave curve:

Graph of f(x) = 2sin(3x - pi/2)

Matched Problem

Problem: Function \( f \) is given by \( f(x) = \dfrac{1}{2}\sin\left(4x + \dfrac{\pi}{2}\right) \).

Find the domain and range of \( f \).
Find the period and the phase shift of the graph of \( f \).
Sketch the graph of function \( f \) over one period.

Graph of Sine, \( a \sin(bx + c) \), Function

Review: Properties of the Basic Sine Function

Graphing \( f(x) = a \sin(bx + c) \)

Examples & Step-by-Step Solutions

More References and Links to Graphing