11.3 Graphs of Trigonometric Functions

Graphs of trigonometric functions are the visual representation of sine, cosine, and tangent. These functions oscillate periodically, creating wave-like patterns that repeat at regular intervals. Understanding their graphs is essential for modeling real-world phenomena like sound waves, tides, and seasonal temperature changes.

Transformations let you modify these graphs by changing their amplitude, period, and position. By applying shifts, stretches, and reflections, you can build functions that accurately model periodic events across science and engineering.

Graphing Trigonometric Functions

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Characteristics of sine, cosine, and tangent functions

Sine: $f(x) = \sin x$ has a period of $2\pi$, an amplitude of 1, and a range of $[-1, 1]$. The graph oscillates smoothly between $-1$ and $1$, crossing the x-axis at multiples of $\pi$ (at $0, \pi, 2\pi$, etc.). It starts at the origin and rises to its maximum of 1 at $x = \frac{\pi}{2}$.

Cosine: $f(x) = \cos x$ also has a period of $2\pi$, an amplitude of 1, and a range of $[-1, 1]$. The difference from sine is where it starts: cosine begins at its maximum value of 1 when $x = 0$. It reaches a minimum of $-1$ at odd multiples of $\pi$ (at $\pi, 3\pi, 5\pi$, etc.) and returns to its maximum at even multiples of $\pi$.

Tangent: $f(x) = \tan x$ has a period of $\pi$, no defined amplitude, and a range of $(-\infty, \infty)$. The graph has vertical asymptotes at odd multiples of $\frac{\pi}{2}$ (at $\frac{\pi}{2}, \frac{3\pi}{2}$, etc.) and crosses the x-axis at multiples of $\pi$. Between each pair of asymptotes, the curve rises from $-\infty$ to $+\infty$.

Key features and midline of trigonometric functions

Every trigonometric graph can be described by four key features:

• Amplitude: the maximum displacement from the midline (applies to sine and cosine; tangent has no amplitude)
• Period: the horizontal length of one complete cycle
• Phase shift: the horizontal translation of the graph (left or right)
• Vertical shift: the vertical translation of the graph (up or down)

The midline is the horizontal line running through the vertical center of the graph. For standard $\sin x$ and $\cos x$, the midline is $y = 0$. When a vertical shift $k$ is applied, the midline moves to $y = k$.

Transformations of Trigonometric Graphs

Types of transformations

Transformations alter the appearance and key features of a trigonometric graph without changing its fundamental wave shape. The general form you'll work with is:

$f(x) = a \sin(b(x - h)) + k \quad \text{or} \quad f(x) = a \cos(b(x - h)) + k$

Vertical translations shift the graph up or down by adding a constant $k$.

• $f(x) = \sin x + 3$ shifts the entire graph up 3 units (midline becomes $y = 3$)
• Positive $k$ shifts up; negative $k$ shifts down

Horizontal translations (phase shifts) shift the graph left or right by subtracting a constant $h$ from the input variable.

• $f(x) = \sin(x - \frac{\pi}{4})$ shifts the graph right by $\frac{\pi}{4}$
• Positive $h$ shifts right; negative $h$ shifts left

Watch the sign carefully here: in $\sin(x - h)$, the subtraction means the shift goes to the right.

Reflections, stretches, and compressions

Reflections flip the graph across an axis.

• Across the x-axis: negate the whole function, e.g., $f(x) = -\sin x$ (peaks become valleys and vice versa)
• Across the y-axis: negate the input, e.g., $f(x) = \sin(-x)$

Vertical stretches and compressions change the amplitude by multiplying the function by a constant $a$.

• $|a| > 1$ stretches the graph vertically (taller waves)
• $|a| < 1$ compresses it vertically (shorter waves)
• Example: $f(x) = 4\cos x$ has an amplitude of 4, so the graph reaches $4$ and $-4$

Horizontal stretches and compressions change the period by multiplying the input variable by a constant $b$.

• $|b| > 1$ compresses the graph horizontally (shorter period, more cycles in the same space)
• $|b| < 1$ stretches it horizontally (longer period, fewer cycles)
• The new period equals $\frac{2\pi}{|b|}$ for sine and cosine, or $\frac{\pi}{|b|}$ for tangent

Combining transformations

Multiple transformations can be applied to a single function. When graphing from the general form $f(x) = a \sin(b(x - h)) + k$, apply them in this order:

1. Reflections (if $a$ is negative)
2. Stretches/compressions (use $|a|$ for vertical, $|b|$ for horizontal)
3. Horizontal translation (phase shift $h$)
4. Vertical translation (shift $k$)

For example, to graph $f(x) = -2\sin(3(x - \frac{\pi}{6})) + 1$:

1. Reflect across the x-axis (negative sign)
2. Amplitude of 2, period of $\frac{2\pi}{3}$
3. Shift right by $\frac{\pi}{6}$
4. Shift up by 1 (midline at $y = 1$)

Modeling Periodic Phenomena

Characteristics of periodic phenomena

Periodic phenomena are events that repeat at regular intervals: sound waves, ocean tides, hours of daylight across the year, or the motion of a Ferris wheel. Trigonometric functions model these well because they naturally repeat.

To set up a model, identify these four values from the context:

• Amplitude: half the distance between the maximum and minimum values. If a tide ranges from 2 ft to 10 ft, the amplitude is $\frac{10 - 2}{2} = 4$ ft.
• Period: the time for one full cycle. If the tide completes a cycle every 12 hours, the period is 12.
• Vertical shift: the average of the max and min values (this becomes the midline). For the tide example: $\frac{10 + 2}{2} = 6$ ft.
• Phase shift: how far the cycle is offset from a standard starting point.

Choosing the appropriate trigonometric function

Pick sine or cosine based on where the cycle begins:

• Use sine if the phenomenon starts at the midline (e.g., a pendulum released from its resting position)
• Use cosine if the phenomenon starts at a maximum or minimum (e.g., a Ferris wheel rider starting at the top)

Then apply transformations to match the data. For the tide example above (max 10 ft, min 2 ft, period 12 hours, starting at max):

$f(t) = 4\cos\!\left(\frac{2\pi}{12}\,t\right) + 6 = 4\cos\!\left(\frac{\pi}{6}\,t\right) + 6$

Once you have the model, you can plug in any time value to predict the tide height, or set the function equal to a specific height and solve for when it occurs.

Amplitude, Period, and Phase Shift

Finding the amplitude

The amplitude is the absolute value of the coefficient in front of the trig function.

For $f(x) = a\sin(bx)$ or $f(x) = a\cos(bx)$, the amplitude is $|a|$.

• $f(x) = 3\sin x$ → amplitude = $|3| = 3$
• $f(x) = -\frac{1}{2}\cos x$ → amplitude = $\left|-\frac{1}{2}\right| = \frac{1}{2}$

Tangent has no amplitude because its range is all real numbers.

Calculating the period

The period depends on the coefficient $b$ multiplying the input variable.

• For sine and cosine: period = $\frac{2\pi}{|b|}$
• For tangent: period = $\frac{\pi}{|b|}$

Examples:

• $f(x) = \cos(2x)$ → period = $\frac{2\pi}{2} = \pi$
• $f(x) = \sin\!\left(\frac{1}{3}x\right)$ → period = $\frac{2\pi}{1/3} = 6\pi$
• $f(x) = \tan(4x)$ → period = $\frac{\pi}{4}$

Determining the phase shift

The phase shift comes from the constant $h$ inside the grouping with $x$. You need the function in the form $\sin(b(x - h))$ to read it correctly.

• $f(x) = \sin(x - \frac{\pi}{4})$ → phase shift = $\frac{\pi}{4}$ to the right
• $f(x) = \cos(x + \frac{\pi}{3})$ → rewrite as $\cos(x - (-\frac{\pi}{3}))$ → phase shift = $\frac{\pi}{3}$ to the left

A common mistake: if the function is written as $\sin(2x - \pi)$, you need to factor out the $b$ first. That's $\sin(2(x - \frac{\pi}{2}))$, so the phase shift is $\frac{\pi}{2}$ to the right, not $\pi$.

Solving problems involving amplitude, period, and phase shift

When a problem gives you information about a trigonometric graph (amplitude, period, phase shift, vertical shift), use it to build the equation:

1. Choose sine or cosine based on the starting behavior
2. Set $|a|$ equal to the amplitude (negate $a$ if the graph is reflected)
3. Solve $\frac{2\pi}{|b|}$ = given period to find $b$
4. Set $h$ equal to the phase shift
5. Set $k$ equal to the vertical shift

In some problems, you may need identities like the Pythagorean identity ($\sin^2 x + \cos^2 x = 1$) or the periodicity property ($\sin(x + 2\pi) = \sin x$) to simplify or verify your work. Always check your answer by confirming that the graph's key points (max, min, zeros, midline crossings) match the given constraints.