Domain and Range - From Graph | How to Find Domain and Range of a ... - Free Printable

The image shows the graphs of two trigonometric functions: \( y = \sin x \) and \( y = \cos x \). Let's analyze and explain the key features of these graphs step by step.

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1. Graph of \( y = \sin x \)

#### Key Features:
- Period: The sine function is periodic with a period of \( 2\pi \). This means the graph repeats itself every \( 2\pi \) units along the \( x \)-axis.
- Amplitude: The amplitude of the sine function is 1. The maximum value of \( y = \sin x \) is 1, and the minimum value is -1.
- Zeros: The sine function crosses the \( x \)-axis (i.e., \( y = 0 \)) at integer multiples of \( \pi \):
\[
x = 0, \pm \pi, \pm 2\pi, \pm 3\pi, \ldots
\]
- Maximum and Minimum Points:
- Maximum points occur at \( x = \frac{\pi}{2} + 2k\pi \) for any integer \( k \).
- Minimum points occur at \( x = \frac{3\pi}{2} + 2k\pi \) for any integer \( k \).
- Behavior:
- The sine function starts at \( (0, 0) \).
- It increases to its maximum value of 1 at \( x = \frac{\pi}{2} \).
- It decreases to its minimum value of -1 at \( x = \frac{3\pi}{2} \).
- It returns to 0 at \( x = 2\pi \), completing one full cycle.

#### Observations from the Graph:
- The graph of \( y = \sin x \) is a smooth wave that oscillates between -1 and 1.
- The function is symmetric about the origin, meaning it is an odd function (\( \sin(-x) = -\sin(x) \)).

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2. Graph of \( y = \cos x \)

#### Key Features:
- Period: The cosine function is also periodic with a period of \( 2\pi \). This means the graph repeats itself every \( 2\pi \) units along the \( x \)-axis.
- Amplitude: The amplitude of the cosine function is 1. The maximum value of \( y = \cos x \) is 1, and the minimum value is -1.
- Zeros: The cosine function crosses the \( x \)-axis (i.e., \( y = 0 \)) at odd multiples of \( \frac{\pi}{2} \):
\[
x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \pm \frac{5\pi}{2}, \ldots
\]
- Maximum and Minimum Points:
- Maximum points occur at \( x = 2k\pi \) for any integer \( k \).
- Minimum points occur at \( x = \pi + 2k\pi \) for any integer \( k \).
- Behavior:
- The cosine function starts at \( (0, 1) \).
- It decreases to its minimum value of -1 at \( x = \pi \).
- It increases back to its maximum value of 1 at \( x = 2\pi \), completing one full cycle.

#### Observations from the Graph:
- The graph of \( y = \cos x \) is a smooth wave that oscillates between -1 and 1.
- The function is symmetric about the \( y \)-axis, meaning it is an even function (\( \cos(-x) = \cos(x) \)).

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Comparison of \( y = \sin x \) and \( y = \cos x \)

1. Phase Shift:
- The graph of \( y = \cos x \) is the same as the graph of \( y = \sin x \), but shifted to the left by \( \frac{\pi}{2} \) units.
- Mathematically, \( \cos x = \sin\left(x + \frac{\pi}{2}\right) \).

2. Symmetry:
- \( y = \sin x \) is an odd function (\( \sin(-x) = -\sin(x) \)).
- \( y = \cos x \) is an even function (\( \cos(-x) = \cos(x) \)).

3. Key Points:
- For \( y = \sin x \):
- \( \sin(0) = 0 \)
- \( \sin\left(\frac{\pi}{2}\right) = 1 \)
- \( \sin(\pi) = 0 \)
- \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
- \( \sin(2\pi) = 0 \)
- For \( y = \cos x \):
- \( \cos(0) = 1 \)
- \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( \cos(\pi) = -1 \)
- \( \cos\left(\frac{3\pi}{2}\right) = 0 \)
- \( \cos(2\pi) = 1 \)

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Final Answer

The graphs illustrate the fundamental properties of the sine and cosine functions, including their periods, amplitudes, zeros, maximum and minimum points, and symmetry. The key takeaway is that:

- \( y = \sin x \) is an odd function with a phase shift relative to \( y = \cos x \).
- \( y = \cos x \) is an even function and can be obtained by shifting \( y = \sin x \) to the left by \( \frac{\pi}{2} \).

\[
\boxed{\text{The graphs show the periodic, oscillatory behavior of } y = \sin x \text{ and } y = \cos x \text{ with periods of } 2\pi.}
\]