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Graphing Trigonometric Functions: A High School Guide

When graphing trigonometric functions, we commonly deal with sine ($\sin$), cosine ($\cos$), and tangent ($\tan$) functions. This guide will focus on sine and cosine since their graphs have similar properties. We’ll cover amplitude, period, phase shift, and vertical shift, with clear examples.

1. Basic Trigonometric Functions

The standard form of the sine and cosine functions is:

Where:

• $A$ is the amplitude.
• $B$ affects the period.
• $C$ causes the phase shift.
• $D$ is the vertical shift.

2. Amplitude

The amplitude measures the height of the wave from the centerline (midline) to the peak. It is given by the absolute value of $A$.

For the sine and cosine functions:

• The default amplitude is 1 (since $\sin$ and $\cos$ range from $-1$ to $1$).
• If $A$ is negative, the graph is reflected across the horizontal axis.

Example: For $y = 3 \sin(x)$, the amplitude is $|A| = 3$. The wave oscillates between $3$ and $-3$.

• Amplitude: $|3| = 3$
• Graph: Taller waves (peaks at 3, valleys at -3)

3. Period

The period of a trigonometric function is the distance (in terms of $x$) it takes for the function to complete one full cycle. The period is affected by $B$ and is calculated as:

Example: For $y = \sin(2x)$, the period is:

The function completes a full cycle from $0$ to $\pi$, making the wave repeat faster.

4. Phase Shift

The phase shift is the horizontal shift of the graph. It is determined by $C$ in the equation $y = A \sin(Bx + C)$ (or $y = A \cos(Bx + C)$). The phase shift can be calculated as:

If the phase shift is positive, the graph moves to the right; if negative, it moves to the left.

Example: For $y = \cos\left(2x - \frac{\pi}{2}\right)$, the phase shift is:

The graph shifts right by $\frac{\pi}{4}$ units.

5. Vertical Shift

The vertical shift is determined by $D$. This shifts the entire graph up or down by $D$ units.

Example: For $y = \sin(x) + 2$, the graph of $y = \sin(x)$ is shifted 2 units up.

• The midline changes from $y = 0$ to $y = 2$.

6. Example 1: Graphing $y = 2 \sin\left(\frac{1}{2}x - \frac{\pi}{3}\right) + 1$

Let’s break this function down:

1. Amplitude: $|A| = 2$. So, the graph oscillates between 1 and 3.

2. Period: $B = \frac{1}{2}$, so the period is:

The graph completes one cycle over a distance of $4\pi$.

3. Phase Shift: The phase shift is:

So, the graph shifts to the right by $\frac{2\pi}{3}$ units.

4. Vertical Shift: $D = 1$. This shifts the graph up by 1 unit, with the midline at $y = 1$.

7. Example 2: Graphing $y = -3 \cos\left(4x + \frac{\pi}{2}\right) - 2$

1. Amplitude: $|A| = 3$. So, the graph oscillates between $-2 + 3 = 1$ and $-2 - 3 = -5$. Since $A$ is negative, the graph is reflected over the x-axis.

2. Period: $B = 4$, so the period is:

The graph completes a cycle in $\frac{\pi}{2}$ units.

3. Phase Shift: The phase shift is:

The graph shifts to the left by $\frac{\pi}{8}$ units.

4. Vertical Shift: $D = -2$. This shifts the graph down by 2 units.

Summary of Key Concepts

• Amplitude: The height from the midline to the peak. $A$ determines how “tall” or “short” the graph is.

• Period: The distance over which the function completes one cycle. It’s affected by $B$ and calculated as $\frac{2\pi}{|B|}$.

• Phase Shift: The horizontal shift of the graph, calculated as $\frac{-C}{B}$. Positive moves right; negative moves left.

• Vertical Shift: The movement of the graph up or down by $D$ units.

These principles help you graph any sine or cosine function by adjusting its amplitude, period, phase shift, and vertical shift.

Let me know if you’d like further clarification or visual examples!