8.1 Graphs of the Sine and Cosine Functions

Sine and cosine functions are the building blocks of periodic motion. These waves show up everywhere, from sound to light to ocean tides. Understanding how to graph and manipulate them is key to modeling real-world phenomena.

This section covers how to adjust sine and cosine graphs using amplitude, frequency, phase shift, and vertical shift. You'll also learn how to read these elements off a graph and write equations from them.

Graphs of Sine and Cosine Functions

Graphing sine and cosine variations

The general form for both functions is:

$y = A \sin(B(x - C)) + D$ and $y = A \cos(B(x - C)) + D$

Each parameter controls a different transformation of the graph:

• $A$ (Amplitude): Determines the vertical stretch or compression.

  • $|A| > 1$ stretches the graph vertically (taller waves)
  • $|A| < 1$ compresses the graph vertically (shorter waves)
  • If $A < 0$, the graph reflects over the x-axis (the wave flips upside down)

• $B$ (Related to frequency): Controls horizontal compression or stretch. A larger $|B|$ squeezes more cycles into the same horizontal space; a smaller $|B|$ spreads cycles out.

• $C$ (Phase shift): Shifts the graph horizontally.

  • $C > 0$ shifts right
  • $C < 0$ shifts left Be careful with signs here. In the form $B(x - C)$, the shift is $C$, not $-C$. So $y = \sin(2(x - \pi))$ shifts right by $\pi$, while $y = \sin(2(x + \pi))$ shifts left by $\pi$ (because $C = -\pi$).

• $D$ (Vertical shift): Moves the entire graph up ($D > 0$) or down ($D < 0$).

Period is the length of one complete cycle. You calculate it with:

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

A few examples to build intuition:

• Standard sine/cosine ($B = 1$): period = $2\pi$
• $B = 2$: period = $\pi$ (the wave completes twice as fast)
• $B = \frac{1}{2}$: period = $4\pi$ (the wave takes twice as long to complete)

Key features of sinusoidal graphs

Midline is the horizontal line the graph oscillates around. It's determined by the vertical shift:

$\text{Midline: } y = D$

For standard sine or cosine, the midline is $y = 0$. If $D = 2$, the midline shifts up to $y = 2$.

Amplitude is the maximum distance from the midline to a peak (or trough). It equals $|A|$. For the standard functions, the amplitude is 1. If $A = 3$, the peaks reach 3 units above the midline and the troughs dip 3 units below it. This means the maximum value of the function is $D + |A|$ and the minimum value is $D - |A|$.

Extrema are the maximum and minimum points. Rather than memorizing formulas for their locations, think about where the parent functions naturally peak and dip, then apply your transformations:

• Standard sine ($y = \sin x$) starts at 0, hits its maximum at $x = \frac{\pi}{2}$, returns to 0 at $x = \pi$, hits its minimum at $x = \frac{3\pi}{2}$, and completes the cycle at $x = 2\pi$.
• Standard cosine ($y = \cos x$) starts at its maximum ($x = 0$), drops to 0 at $x = \frac{\pi}{2}$, hits its minimum at $x = \pi$, returns to 0 at $x = \frac{3\pi}{2}$, and completes the cycle at $x = 2\pi$.

To find extrema for a transformed function, take the standard locations, divide by $|B|$ (to account for the period change), then add $C$ (for the phase shift). The y-values of the extrema are $D + |A|$ (max) and $D - |A|$ (min).

Equations from sinusoidal graphs

When you're given a graph and need to write the equation, follow these steps:

1. Find the midline ($D$). Average the maximum and minimum y-values: $D = \frac{y_{\text{max}} + y_{\text{min}}}{2}$

2. Find the amplitude ($A$). Measure the distance from the midline to a peak: $A = y_{\text{max}} - D$

3. Find the period ($P$) and calculate $B$. Measure the horizontal distance for one full cycle (peak to next peak, or any point to the next identical point). Then: $B = \frac{2\pi}{P}$

4. Find the phase shift ($C$). Compare the graph to the parent function. If you're writing a cosine equation, look at where the nearest maximum occurs; that x-value is your $C$. If you're writing a sine equation, look at where the graph crosses the midline going upward; that x-value is your $C$.

5. Plug everything into the general form: $y = A \sin(B(x - C)) + D$ or $y = A \cos(B(x - C)) + D$

Worked example: Suppose a tide graph has a maximum height of 6 m, a minimum height of 2 m, completes one cycle every 12 hours, and the first high tide occurs at $x = 3$ hours.

• Midline: $D = \frac{6 + 2}{2} = 4$
• Amplitude: $A = 6 - 4 = 2$
• Period: $P = 12$, so $B = \frac{2\pi}{12} = \frac{\pi}{6}$
• Phase shift: The first maximum is at $x = 3$, and cosine naturally peaks at $x = 0$, so $C = 3$

The equation is: $y = 2\cos\left(\frac{\pi}{6}(x - 3)\right) + 4$

You could also write this as a sine function with a different phase shift, since the sine's first upward midline crossing would occur at a different x-value. Both forms are valid.

Additional Concepts

• Periodic functions repeat their values at regular intervals. Sine and cosine are the most common examples, but any function where $f(x + P) = f(x)$ for some constant $P$ is periodic.
• The unit circle is the foundation for understanding why sine and cosine behave the way they do. The x-coordinate of a point on the unit circle gives cosine, and the y-coordinate gives sine.
• You can always convert between sine and cosine equations for the same graph. Since $\cos(x) = \sin\left(x + \frac{\pi}{2}\right)$, a cosine function is just a sine function shifted left by $\frac{\pi}{2}$.