Lesson 8.1 : Graphs of the Sine and Cosine Functions

New Concept

Sine and cosine functions graph as periodic waves. We'll explore how to transform these waves by adjusting their amplitude ($A$), period ($P = \frac{2\pi}{|B|}$), phase shift, and vertical shift using the general forms $y = A \operatorname{sin}(Bx-C)+D$ and $y = A \operatorname{cos}(Bx-C)+D$.

What’s next

Now, let's apply this. You'll work through interactive examples to identify transformations and then create graphs in a series of practice cards and challenges.

Property

The sine and cosine functions have several distinct characteristics:

• They are periodic functions with a period of $2\pi$.
• The domain of each function is $(-\infty, \infty)$ and the range is $[-1, 1]$.
• The graph of $y = \operatorname{sin}(x)$ is symmetric about the origin, because it is an odd function.
• The graph of $y = \operatorname{cos}(x)$ is symmetric about the $y$-axis, because it is an even function.

A periodic function is a function for which a specific horizontal shift, $P$, results in a function equal to the original function: $f(x + P) = f(x)$ for all values of $x$ in the domain of $f$. When this occurs, we call the smallest such horizontal shift with $P > 0$ the period of the function.

Examples

• The function $f(x) = \operatorname{cos}(x)$ repeats its values every $2\pi$ units, so its period is $2\pi$. Since its graph is a mirror image across the y-axis, it is an even function.
• The function $f(x) = \operatorname{sin}(x)$ has a domain of all real numbers and a range of $[-1, 1]$. This means it can take any x-input, but its output will always be between -1 and 1, inclusive.
• The graph of $y = \operatorname{sin}(x)$ is an odd function, which means that $\operatorname{sin}(-x) = -\operatorname{sin}(x)$. For instance, $\operatorname{sin}(-\frac{\pi}{2}) = -1$, which is the same as $-\operatorname{sin}(\frac{\pi}{2})$.

Explanation

These are the foundational rules for sine and cosine waves. Their repeating nature (period), their height limits (range), and their specific symmetries (odd/even) make their behavior predictable and useful for modeling cycles.

Property

If we let $C = 0$ and $D = 0$ in the general form equations of the sine and cosine functions, we obtain the forms

The period is $\dfrac{2\pi}{|B|}$.

Examples

• To find the period of $f(x) = \operatorname{sin}(4x)$, we use the formula $P = \frac{2\pi}{|B|}$. Here, $B=4$, so the period is $P = \frac{2\pi}{4} = \frac{\pi}{2}$.
• Determine the period of the function $g(x) = \operatorname{cos}(\frac{\pi}{5}x)$. In this case, $B = \frac{\pi}{5}$, so the period is $P = \frac{2\pi}{|\frac{\pi}{5}|} = 2\pi \cdot \frac{5}{\pi} = 10$.
• For the function $h(x) = \operatorname{sin}(\frac{1}{3}x)$, the value of $B$ is $\frac{1}{3}$. The period is $P = \frac{2\pi}{|\frac{1}{3}|} = 6\pi$. This graph is stretched horizontally.

Explanation

The period is the horizontal length of one full cycle of the wave. The value of $|B|$ in the equation determines this length. A larger $|B|$ compresses the wave horizontally, resulting in a shorter period.

Property

If we let $C = 0$ and $D = 0$ in the general form equations of the sine and cosine functions, we obtain the forms

The amplitude is $|A|$, which is the vertical height from the midline. In addition, notice in the example that

Examples

• What is the amplitude of the function $f(x) = -5\operatorname{sin}(x)$? Here, $A = -5$, so the amplitude is $|A| = |-5| = 5$. The negative sign reflects the graph over the x-axis.
• For the function $g(x) = \frac{2}{3}\operatorname{cos}(x)$, the amplitude is $|A| = |\frac{2}{3}| = \frac{2}{3}$. This means the function's graph is vertically compressed.
• A sinusoidal function has a maximum value of 9 and a minimum value of 1. Its amplitude is calculated as $|A| = \frac{1}{2}|9 - 1| = \frac{1}{2}(8) = 4$.

Explanation

Amplitude measures the wave's height from its center line (midline) to its peak. The value $|A|$ tells you this maximum distance. A negative sign on A reflects the graph across the midline without changing the amplitude.

Property

Given an equation in the form $f(x) = A \operatorname{sin}(Bx - C) + D$ or $f(x) = A \operatorname{cos}(Bx - C) + D$, $\dfrac{C}{B}$ is the phase shift and $D$ is the vertical shift.
To identify the characteristics of a sinusoidal function:

1. Determine the amplitude as $|A|$.
2. Determine the period as $P = \dfrac{2\pi}{|B|}$.
3. Determine the phase shift as $\dfrac{C}{B}$.
4. Determine the midline as $y = D$.

Examples

• In the function $f(x) = \operatorname{cos}(x - \frac{\pi}{4}) + 3$, the phase shift is $\frac{C}{B} = \frac{\pi/4}{1} = \frac{\pi}{4}$ (to the right), and the vertical shift is $D=3$ (up). The midline is $y=3$.
• To find the phase shift for $g(x) = \operatorname{sin}(2x + \pi)$, we rewrite it as $g(x) = \operatorname{sin}(2x - (-\pi))$. Here $C=-\pi$ and $B=2$, so the phase shift is $\frac{-\pi}{2}$, meaning $\frac{\pi}{2}$ units to the left.
• Analyze $y = 5\operatorname{cos}(\pi x - 2\pi) - 1$. The amplitude is $|A|=5$. The period is $P = \frac{2\pi}{|\pi|} = 2$. The phase shift is $\frac{C}{B} = \frac{2\pi}{\pi} = 2$ (2 units right). The midline is $y=-1$.

Explanation

The phase shift, $\frac{C}{B}$, slides the graph horizontally (left or right). The vertical shift, $D$, moves the entire graph vertically, establishing a new midline for the wave at $y=D$, which is the new horizontal center line.