Question

a) State the five key points for \(y=\cos x\) that occur in one complete cycle from 0 to \(2 \pi\) b) Use the key points to sketch a graph of \(y=\cos x\) for \(-2 \pi \leq x \leq 2 \pi .\) Indicate the key points on your graph. c) What are the \(x\) -intercepts of the graph? d) What is the \(y\) -intercept of the graph? e) What is the maximum value of the graph? the minimum value?

Short answer

Key points are at 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), 2 \( 2\pi \). The x-intercepts are at \( -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \). The y-intercept is at (0,1). The maximum value is 1, and the minimum value is -1.

Step-by-step solution

State the Five Key Points

For the function \( y = \cos x \), the key points in one complete cycle from 0 to \( 2 \pi \) occur at \(x = 0 \), \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). At these points, the values of \( y = \cos x \) are 1, 0, -1, 0, and 1, respectively.

Sketch the Graph and Key Points

To sketch the graph of \( y = \cos x \) for \( -2 \pi \leq x \leq 2 \pi \), plot the key points found in the previous step for one cycle and replicate this pattern for the range. The key points are: \((-2\pi, 1)\), \((-\frac{3\pi}{2}, 0)\), \((-\pi, -1)\), \((-\frac{\pi}{2}, 0)\), \((0, 1)\), \((\frac{\pi}{2}, 0)\), \((\pi, -1)\), \((\frac{3\pi}{2}, 0)\), and \((2\pi, 1)\). Connect these points with a smooth curve to get the graph.

Determine the x-Intercepts

The \( x \)-intercepts, where the graph crosses the x-axis, are the points where \( \cos x = 0 \). These occur at \( x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \) within the given range.

Determine the y-Intercept

The \( y \)-intercept is the point where the graph crosses the y-axis. For \( y = \cos x \), this occurs at \( x = 0 \), so the \( y \)-intercept is the point \( (0, 1) \).

Determine the Maximum and Minimum Values

The maximum value of \( \cos x \) is 1, which occurs at \( x = 2k\pi \) for any integer \( k \). The minimum value of \( \cos x \) is -1, which occurs at \( x = (2k+1)\pi \) for any integer \( k \). Within the given range, the maximum value is 1 and the minimum value is -1.

Key concepts

Key Points of Cosine Function

The cosine function, denoted as \(y=\cos x\), has specific key points that occur in one complete cycle from 0 to \(2\pi\). These key points are essential for understanding how the function behaves over its period.
For \(y=\cos x\), the key points occur at:

• \(x=0\)
• \(x=\frac{\pi}{2}\)
• \(x=\pi\)
• \(x=\frac{3\pi}{2}\)
• \(x=2\pi\)

At these points, the corresponding values of \(y=\cos x\) are:

• 1
• 0
• -1
• 0
• 1

Understanding these key points helps to recognize the periodical nature of the cosine function and forms the basis for graphing it effectively.

Graphing Trigonometric Functions

Graphing trigonometric functions like \(y=\cos x\) involves plotting the key points and then drawing a smooth curve. To sketch the graph of the function for \(-2\pi \leq x \leq 2\pi\), follow these steps:

• Plot the key points from one complete cycle: \((0,1)\), \((\frac{\pi}{2},0)\), \((\pi,-1)\), \((\frac{3\pi}{2},0)\), and \((2\pi,1)\).
• Replicate the pattern for the given range past these initial points: \((-2\pi,1)\), \((-\frac{3\pi}{2},0)\), \((-\pi,-1)\), \((-\frac{\pi}{2},0)\), and \((0,1)\).

Connecting these points with a smooth wave-like curve will give you the graph. Understanding how to plot these can visually demonstrate how the \(\cos x\) function behaves over its period, alternating between its maximum and minimum values.

Trigonometric Intercepts

Intercepts are points where the graph crosses the axes. For the function \(y=\cos x\), both x- and y-intercepts are significant.
The x-intercepts are the points where the graph crosses the x-axis, which happens when \(\cos x = 0\). For the range \(-2\pi \leq x \leq 2\pi\), this occurs at:

• \(-\frac{3\pi}{2}\)
• \(-\frac{\pi}{2}\)
• \((\frac{\pi}{2}\)
• \((\frac{3\pi}{2}\)

These are the points where the graph touches the x-axis.

The y-intercept is the point where the graph crosses the y-axis. For \(y=\cos x\), this occurs at \(x=0\). Hence, the y-intercept is \((0,1)\). Intercepts are helpful in plotting the graph and understanding where the function equals zero or intersects the axes.

Maximum and Minimum Values in Trigonometry

In trigonometry, the maximum and minimum values of functions tell us about their range and behavior.
For the cosine function \(y=\cos x\), these extreme values are crucial.
The maximum value of \(\cos x\) is:\[y=1\]
This occurs at points where \(x=2k\pi\) for any integer \(k\).
The minimum value of \(\cos x\) is:\[y=-1\]
This happens at points where \(x=(2k+1)\pi\) for any integer \(k\).
Within the range \(-2\pi \leq x \leq 2\pi\), the function oscillates between these maximum and minimum values, defining its amplitude. Understanding the maxima and minima is vital for analyzing the complete behavior of the trigonometric functions.