Amplitude And Period Of Trigonometric Functions

What Are Amplitude and Period in Trigonometry?

The amplitude of a sinusoidal function (like sine or cosine) is its maximum distance from the function's midline or center line. In simpler terms, it's the height of the wave from the center. A larger amplitude means a taller wave, while a smaller amplitude means a shorter one. Since amplitude is a distance, it is always a positive value.

The period of a sinusoidal function is the length of one complete cycle of the graph before it starts repeating. It's the horizontal distance measured along the x-axis from one point on the wave to the next identical point, for example, from one crest to the next. The period tells us how frequently the wave pattern repeats.

For the basic functions we'll be discussing, like $y=A\sin (Bx)$, the midline is the x-axis (the line $y=0$). The amplitude is the distance from this axis to the highest point (the crest) or the lowest point (the trough).

A Quick Look at the Basic Sine and Cosine Graphs

Before we start stretching and squeezing these graphs, we need to be familiar with the parent functions: $y=\sin (x)$ and $y=\cos (x)$. Both of these functions are periodic, meaning their graphs repeat in a predictable cycle.

• $y=\sin (x)$: This graph starts at the origin $(0,0)$, rises to a maximum of $1$ at $x=\pi /2$, returns to zero at $x=\pi$, drops to a minimum of $-1$ at $x=3\pi /2$, and completes its cycle back at zero at $x=2\pi$.
• $y=\cos (x)$: This graph starts at its maximum value of $1$ at $x=0$, crosses the x-axis at $x=\pi /2$, hits its minimum of $-1$ at $x=\pi$, crosses the axis again at $x=3\pi /2$, and returns to its maximum to complete the cycle at $x=2\pi$.

For both of these parent functions, the amplitude is $1$ (since the graph goes from $-1$ to $1$), and the period is $2\pi$ (the length of one full cycle). Here is a quick comparison:

How Does the 'A' Value Control Amplitude?

When we introduce a coefficient in front of our trigonometric function, we create a vertical stretch or compression. The general form is $y=A\sin (Bx)$ or $y=A\cos (Bx)$. The value of $A$ directly controls the amplitude.

The formula for amplitude is simple and direct:

The absolute value is crucial because amplitude represents a distance, which can never be negative.

• If $|A|>1$, the graph is stretched vertically, making the waves taller.
• If $0<|A|<1$, the graph is compressed vertically, making the waves shorter.
• If $A$ is negative, the graph is reflected across the x-axis. The amplitude is still the positive value $|A|$, but the shape of the graph is flipped. For example, $y=-\cos (x)$ starts at its minimum value instead of its maximum.

How Does the 'B' Value Control the Period?

The coefficient $B$ inside the trigonometric function, next to the $x$, controls the horizontal stretching or compression of the graph. This, in turn, changes the period of the function. The value $B$ is sometimes called the frequency.

The relationship is inverse: a larger value of $B$ makes the graph complete its cycle faster, resulting in a shorter period. A smaller value of $B$ (between $0$ and $1$) slows the cycle down, resulting in a longer period.

The formula to calculate the period $P$ for sine and cosine is:

Think of it this way: the standard cycle for $\sin (\theta )$ completes when $\theta$ goes from $0$ to $2\pi$. For our function $\sin (Bx)$, the cycle completes when the input $Bx$ goes from $0$ to $2\pi$. By solving $Bx=2\pi$ for $x$, we get $x=\frac{2\pi }{B}$, which is the length of the period.

Putting It All Together: Finding Amplitude and Period from an Equation

Now we can combine these two concepts to analyze any function in the form $y=A\sin (Bx)$ or $y=A\cos (Bx)$. The process is straightforward: identify $A$ and $B$, then apply the respective formulas.

1. Identify $A$: This is the coefficient in front of the sine or cosine function.
2. Calculate Amplitude: Use the formula Amplitude = $|A|$. Note whether $A$ is negative, as this indicates a reflection.
3. Identify $B$: This is the coefficient multiplying $x$ inside the function.
4. Calculate Period: Use the formula Period = $\frac{2\pi }{|B|}$.

This analysis gives you the two most important characteristics for graphing the function.

How to Graph Sine and Cosine with a New Amplitude and Period

Once you know the amplitude and period, you can sketch an accurate graph of the function. A helpful technique is to find the five key points of one cycle.

1. Find the Amplitude and Period: First, calculate Amplitude = $|A|$ and Period $P=\frac{2\pi }{|B|}$. This will define the boundaries of your graph. The y-values will be between $-|A|$ and $|A|$, and one cycle will have a length of $P$ on the x-axis.
2. Find the Key Point Intervals: Divide the period by 4 to get the spacing of your key points: $\frac{P}{4}$. The five key points for the first cycle starting at $x=0$ will be at $x=0,\frac{P}{4},\frac{P}{2},\frac{3P}{4},P$.
3. Determine the Pattern:
   • For a sine function $y=A\sin (Bx)$, the pattern is: midline, maximum, midline, minimum, midline. If $A$ is negative, the pattern is: midline, minimum, midline, maximum, midline.
   • For a cosine function $y=A\cos (Bx)$, the pattern is: maximum, midline, minimum, midline, maximum. If $A$ is negative, the pattern is: minimum, midline, maximum, midline, minimum.
4. Plot and Connect: Plot the five key points and connect them with a smooth, rounded curve. You can then extend the pattern in both directions to draw more cycles.

For instance, to graph $y=2\sin (\pi x)$: The amplitude is $2$. The period is $\frac{2\pi }{\pi }=2$. The key points are at $x=0,0.5,1,1.5,2$. Since it's a positive sine function, the y-values at these points will be $0,2,0,-2,0$. Plot these five points and draw the wave.

Common Mistakes to Avoid

When working with amplitude and period, a few common errors can trip students up. Be on the lookout for these:

• Reporting a Negative Amplitude: Amplitude is a distance and must always be positive. The amplitude of $y=-3\sin (x)$ is $3$, not $-3$. The negative sign simply indicates a reflection over the x-axis.
• Confusing B with the Period: Students sometimes mistakenly think that the value of $B$ is the period. Remember, $B$ is the frequency used in the formula $P=\frac{2\pi }{|B|}$ to calculate the period.
• Using the Incorrect Period Formula: A common algebraic mistake is to multiply by $B$ instead of dividing. The formula is $\frac{2\pi }{|B|}$, not $2\pi |B|$. For the tangent function, the formula is different altogether ($P=\frac{\pi }{|B|}$).
• Forgetting the Reflection: When $A$ is negative, it's easy to calculate the amplitude correctly but then forget to flip the graph. A reflected cosine graph starts at its minimum, not its maximum. A reflected sine graph goes down first, not up.

Quick Reference Summary

For any trigonometric function in the form $y=A\sin (Bx)$ or $y=A\cos (Bx)$:

• Amplitude: The amplitude measures the vertical height of the waves from the midline. It is calculated with the formula:

  Amplitude = $|A|$

• Period: The period measures the horizontal length of one complete cycle of the graph. It is calculated with the formula:

  Period = $\frac{2\pi }{|B|}$

Remember that a negative value for $A$ reflects the graph across the x-axis, but does not change the amplitude itself.