Question

Sketch the graph of each function over the interval \(\left[-360^{\circ}, 360^{\circ}\right] .\) For each function, clearly label the maximum and minimum values, the \(x\) -intercepts, the \(y\) -intercept, the period, and the range. a) \(y=2 \cos x\) b) \(y=-3 \sin x\) c) \(y=\frac{1}{2} \sin x\) d) \(y=-\frac{3}{4} \cos x\)

Short answer

Graphs have respective amplitudes, periods of 360^{\circ}, and range values: (-2,2), (-3,0), (-0.5,0.5), (-0.75,0). Maximum and minimum values, intercepts specified.

Step-by-step solution

Identify Key Characteristics

For each function, we need to identify the key features including maximum and minimum values, x-intercepts, y-intercept, period, and range.

Sketch Graph of y = 2\cos x

The function is a cosine wave with amplitude 2. Maximum value: 2, Minimum value: -2, x-intercepts at \(90^{\circ}, 270^{\circ}, -90^{\circ}, -270^{\circ}\), y-intercept: 2, Period: 360^{\circ}, Range: [-2, 2].

Sketch Graph of -3\sin x

The function is a sine wave with amplitude 3, vertically flipped. Maximum value: 0, Minimum value: -3, x-intercepts at \(0^{\circ}, 180^{\circ}, -180^{\circ}, 360^{\circ}, -360^{\circ}\), y-intercept: 0, Period: 360^{\circ}, Range: [-3, 0].

Sketch Graph of \frac{1}{2}\sin x

This is a sine wave with reduced amplitude of 0.5. Maximum value: 0.5, Minimum value: -0.5, x-intercepts at \(0^{\circ}, 180^{\circ}, -180^{\circ}, 360^{\circ}, -360^{\circ}\), y-intercept: 0, Period: 360^{\circ}, Range: [-0.5, 0.5].

Sketch Graph of -\frac{3}{4}\cos x

This is a cosine wave with amplitude of 0.75, vertically flipped. Maximum value: 0, Minimum value: -0.75, x-intercepts at \(90^{\circ}, 270^{\circ}, -90^{\circ}, -270^{\circ}\), y-intercept: -0.75, Period: 360^{\circ}, Range: [-0.75, 0].

Key concepts

cosine wave

A cosine wave is a type of trigonometric function specifically defined by the wave equation of the form:
\[y = A \cos(Bx + C) + D\]
Here, A stands for the amplitude, B for the frequency, C for the phase shift, and D for the vertical shift.
The standard cosine wave has an amplitude of 1, a period of 360 degrees, and it is symmetric about the y-axis.

• The highest points on the wave are called maxima.
• The lowest points are called minima.
• It starts from its maximum value when x = 0 (y-intercept).

For instance, in the function \(y = 2 \cos x\), the wave has an amplitude of 2, causing it to oscillate between +2 and -2.

sine wave

A sine wave is another fundamental trigonometric function, typically expressed by:
\[y = A \sin(Bx + C) + D\]
Like the cosine wave, A is the amplitude, B represents the frequency, C is the phase shift, and D indicates the vertical shift.
The sine wave also features a period of 360 degrees in its standard form and oscillates about the x-axis.

• The maximum points are the peaks of the wave.
• The minima are the troughs of the wave.
• It intersects the origin (0,0), which is the y-intercept in the standard sine wave.

For example, the function \(y = -3 \sin x\) has an amplitude of 3 and is reflected over the x-axis due to the negative sign.

amplitude

Amplitude measures the height of the wave from its central axis to its peak or trough. It is denoted by the absolute value of A in the sine and cosine equations.

• In \(y = 2 \cos x\), the amplitude is 2, meaning the wave reaches 2 units above and below its midline.
• Similarly, in \(y = \frac{1}{2} \sin x\), the amplitude is 0.5, resulting in a much smaller oscillation.

Amplitude reflects the energy or intensity of the wave. The larger the amplitude, the more pronounced the wave's peaks and troughs are.

period

The period of a wave is the distance it takes for the wave to complete one full cycle. For sine and cosine waves, this can be determined using the formula:
\[\text{Period} = \frac{360^{\circ}}{B}\]
Here, B is the coefficient of x.

• For the function \(y = 2 \cos x\), the period is \(\frac{360^{\circ}}{1} = 360^{\circ}\).
• Similarly, for \(y = -3 \sin x\), the period remains \(360^{\circ}\) as well.

In essence, a period indicates how long it takes for the wave to reset its starting point and can greatly affect the appearance of the graph.

x-intercepts

X-intercepts are the points where the function crosses the x-axis. For trig functions, these are the points where the value of the function is zero.

• For the cosine function \(y = 2 \cos x\), x-intercepts occur at 90° increments: 90°, 270°, -90°, -270°.
• For the sine function \(y = -3 \sin x\), x-intercepts occur at 180° intervals: 0°, 180°, -180°, 360°, -360°.

These intercepts are crucial for sketching the basic shape of sine and cosine waves, as they delineate the points where the wave crosses the axis and changes its direction.