Graphs Of Trigonometric Functions

What Are Trigonometric Graphs?

A trigonometric graph is a visual representation of a trigonometric function, such as sine, cosine, or tangent, plotted on the Cartesian coordinate plane. These graphs show the relationship between an angle (plotted on the x-axis, typically in radians) and the value of the trigonometric function at that angle (plotted on the y-axis). The fundamental idea is to plot points of the form $(x,f(x))$, where $f(x)$ is a function like $\sin (x)$ or $\cos (x)$.

Unlike linear or quadratic graphs, trigonometric graphs are periodic, meaning their shape repeats at regular intervals. This cyclical nature comes directly from the unit circle. As we travel around the unit circle, the values for sine (the y-coordinate) and cosine (the x-coordinate) repeat every $2\pi$ radians, or ${360}^{\circ }$. This repetition creates the characteristic wave-like patterns, known as sinusoidal waves for sine and cosine, that are central to their study.

Exploring the Parent Sine and Cosine Functions

The two most fundamental trigonometric graphs are $y=\sin (x)$ and $y=\cos (x)$. These are often called the 'parent functions' because all other transformations—stretching, shifting, and compressing—start from these basic shapes.

The Sine Function: $y=\sin (x)$

The sine graph is a smooth, continuous wave that passes through the origin $(0,0)$. We can plot it by using key points from one full rotation of the unit circle, from $x=0$ to $x=2\pi$.

• At $x=0$, $\sin (0)=0$.
• At $x=\pi /2$, $\sin (\pi /2)=1$ (the maximum value).
• At $x=\pi$, $\sin (\pi )=0$.
• At $x=3\pi /2$, $\sin (3\pi /2)=-1$ (the minimum value).
• At $x=2\pi$, $\sin (2\pi )=0$ (the cycle completes).

Key Properties of $y=\sin (x)$:

• Domain: All real numbers, $(-\mathrm{\infty },\mathrm{\infty })$.
• Range: $[-1,1]$. The graph never goes above $1$ or below $-1$.
• Period: $2\pi$. The graph completes one full cycle every $2\pi$ radians.
• Amplitude: $1$. The amplitude is half the distance between the maximum and minimum values, which is $(1-(-1))/2=1$.

The Cosine Function: $y=\cos (x)$

The cosine graph looks very similar to the sine graph. In fact, it's the same wave, just shifted horizontally. The cosine graph starts at its maximum value at $x=0$ because $\cos (0)=1$.

• At $x=0$, $\cos (0)=1$ (the maximum value).
• At $x=\pi /2$, $\cos (\pi /2)=0$.
• At $x=\pi$, $\cos (\pi )=-1$ (the minimum value).
• At $x=3\pi /2$, $\cos (3\pi /2)=0$.
• At $x=2\pi$, $\cos (2\pi )=1$ (the cycle completes).

Key Properties of $y=\cos (x)$:

• Domain: All real numbers, $(-\mathrm{\infty },\mathrm{\infty })$.
• Range: $[-1,1]$.
• Period: $2\pi$.
• Amplitude: $1$.

An important observation is that the graph of $y=\cos (x)$ is identical to the graph of $y=\sin (x+\pi /2)$. This means the cosine function is simply the sine function shifted $\pi /2$ units to the left.

How Do Amplitude and Period Transform the Graphs?

Once we understand the parent functions, we can start transforming them. The general form we'll explore is $y=A\sin (Bx)$ and $y=A\cos (Bx)$. The coefficients $A$ and $B$ control the vertical and horizontal stretching of the graph, respectively.

Amplitude: The $A$ Value

The amplitude measures the height of the wave from its central axis. It is determined by the coefficient $A$ in front of the function.

The amplitude is the absolute value of $A$. It tells you the maximum distance the graph reaches from its horizontal midline. If $A$ is negative, the graph is also reflected across the x-axis, but the amplitude itself remains positive. For example, both $y=3\cos (x)$ and $y=-3\cos (x)$ have an amplitude of $3$.

Period: The $B$ Value

The period is the length of one complete cycle of the graph. It is determined by the coefficient $B$ of the variable $x$.

The value of $B$ controls the horizontal compression or stretching of the wave. A value of $|B|>1$ compresses the graph, making the period shorter (more cycles fit into a given interval). A value of $|B|<1$ stretches the graph, making the period longer.

What Are Phase Shift and Vertical Shift?

Now we introduce horizontal and vertical translations to our functions. The complete form for a sinusoidal function is $y=A\sin (B(x-C))+D$ or $y=A\cos (B(x-C))+D$.

Vertical Shift: The $D$ Value

The vertical shift moves the entire graph up or down. The value of $D$ determines the new horizontal midline of the graph.

If $D$ is positive, the graph shifts up by $D$ units. If $D$ is negative, it shifts down. The new midline of the graph is the line $y=D$. The range of the function becomes $[D-|A|,D+|A|]$.

Phase Shift: The $C$ Value

The phase shift is the horizontal translation of the graph. It tells you where the starting point of the cycle moves to.

Important: To correctly identify $C$, the function must be in the form $B(x-C)$. If you have an expression like $\sin (2x-\pi )$, you must first factor out the $B$ value: $\sin (2(x-\pi /2))$. Here, the phase shift $C$ is $\pi /2$, not $\pi$. A positive $C$ value shifts the graph to the right, and a negative $C$ value (e.g., in $x+C=x-(-C))$ shifts it to the left.

How Can We Graph Any Sine or Cosine Function?

Graphing a complex trigonometric function can be broken down into a reliable, step-by-step process. Let's use the general form $y=A\sin (B(x-C))+D$.

1. Rewrite and Identify: Ensure the function is in the standard form by factoring out $B$ if necessary. Then, identify the values of $A,B,C,$ and $D$.
2. Determine Key Properties: Calculate the amplitude $(|A|)$, period $(2\pi /|B|)$, phase shift $(C)$, and vertical shift $(D)$. Note if there is a reflection (if $A<0$).
3. Establish the Frame: Draw the midline, $y=D$. Then, draw the upper and lower boundaries at $y=D+|A|$ and $y=D-|A|$. This creates a vertical 'frame' for your graph.
4. Find the Cycle Interval: The cycle starts at $x=C$ and ends at $x=C+\text{Period}$. Mark these on the x-axis. This is the horizontal 'frame'.
5. Mark Quarter Points: Divide the cycle interval into four equal subintervals. These five x-values (start, quarter, half, three-quarter, end) are where the maximums, minimums, and midline points will occur.
6. Plot and Sketch: Plot the five key points for one cycle, remembering the basic shape of sine (mid-max-mid-min-mid) or cosine (max-mid-min-mid-max). If $A$ is negative, reflect the pattern across the midline (e.g., sine becomes mid-min-mid-max-mid). Connect the points with a smooth curve.

What Does the Tangent Graph Look Like?

The graph of the tangent function, $y=\tan (x)$, is very different from sine and cosine. Since $\tan (x)=\frac{\sin (x)}{\cos (x)}$, its behavior is dictated by both functions.

Key features arise from the denominator, $\cos (x)$. Whenever $\cos (x)=0$, the tangent function is undefined, resulting in a vertical asymptote. This occurs at $x=\dots ,-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2},\dots$ or more generally at $x=\frac{\pi }{2}+n\pi$ for any integer $n$.

Between these asymptotes, the graph is a rising curve that passes through the origin. Unlike sine and cosine, the tangent function's range is all real numbers.

Key Properties of $y=\tan (x)$:

• Domain: All real numbers except $x=\frac{\pi }{2}+n\pi$.
• Range: All real numbers, $(-\mathrm{\infty },\mathrm{\infty })$.
• Period: $\pi$. The repeating pattern is shorter than for sine and cosine.
• Asymptotes: Vertical lines at each value where the function is undefined.
• Zeros: The graph crosses the x-axis whenever $\sin (x)=0$, which is at $x=n\pi$.

To graph one cycle of $y=\tan (x)$, we typically plot the section between the asymptotes at $x=-\frac{\pi }{2}$ and $x=\frac{\pi }{2}$. Key points in this cycle include:

• $x=-\pi /4,y=-1$
• $x=0,y=0$
• $x=\pi /4,y=1$

The transformed tangent function $y=A\tan (B(x-C))+D$ follows similar rules. The value $A$ causes a vertical stretch, and $D$ and $C$ cause vertical and phase shifts. The period of the transformed tangent function is given by $\frac{\pi }{|B|}$.

What Are Common Mistakes When Graphing Trig Functions?

Graphing trigonometric functions involves many steps, and it's easy to make small errors. Here are some of the most common mistakes to watch out for:

• Incorrect Phase Shift: For a function like $y=\sin (2x-\pi )$, many students mistakenly identify the phase shift as $\pi$. You MUST factor out the $B$ value first to get $y=\sin (2(x-\pi /2))$. The correct phase shift is $\pi /2$ to the right.
• Confusing Period Formulas: Remember that the period for sine and cosine is $2\pi /|B|$, but the period for tangent is $\pi /|B|$. Using the wrong formula will lead to an incorrect horizontal stretch or compression.
• Mixing Radians and Degrees: The x-axis on these graphs is almost always scaled in radians. Ensure your calculations for key points are also in radians. Trying to plot a point at $x=90$ instead of $x=\pi /2$ will produce a very wrong graph.
• Forgetting Reflections: A negative value for $A$ (e.g., $y=-2\cos (x)$) reflects the entire graph across its midline. A common mistake is to calculate the amplitude correctly as $2$ but forget to flip the graph's pattern (e.g., plotting a cosine wave that starts at its maximum instead of its minimum).
• Misplacing the Starting Point: Remember that a basic sine curve starts at its midline, while a basic cosine curve starts at its maximum. Applying a phase shift moves this starting point, but the initial shape relative to that starting point remains the same (unless reflected).

Quick Summary of Graph Properties

This table provides a quick reference for the properties of the sinusoidal function $y=Af(B(x-C))+D$, where $f$ is either $\sin$ or $\cos$.