Understanding Amplitude and Period of Trigonometric Functions

The video tutorial explains how to determine the amplitude and period of the function y = -1/2 cos(3x). It begins by defining amplitude as half the difference between the maximum and minimum values of a periodic function. The amplitude is calculated as the absolute value of the coefficient of the cosine function, which is 1/2. The period is defined as the length of the smallest interval containing one complete cycle of the function. The period is calculated by dividing 2 pi by the absolute value of the coefficient of x, resulting in 2 pi/3. The tutorial concludes with a visualization of the function's amplitude and period on a graph.

What is the amplitude of a periodic function?

• The maximum value it reaches

• The minimum value it reaches

• Half the difference between its maximum and minimum values

• The average of its maximum and minimum values

Why do we use the absolute value when determining the amplitude of y = -1/2 cos(3x)?

• To adjust for the period

• To simplify the calculation

• To account for any shifts in the graph

• To ensure the amplitude is always positive

What does the period of a periodic function represent?

• The length of the smallest interval that contains one complete cycle

• The average value over one cycle

• The time it takes to reach its maximum value

• The distance between two peaks

How do you calculate the period of y = -1/2 cos(3x)?

• Add 2π to the coefficient of x

• Divide 2π by the coefficient of x

• Multiply 2π by the coefficient of x

• Subtract 2π from the coefficient of x

Why does the coefficient of x affect the period of a trigonometric function?

• It affects the vertical stretch

• It changes the amplitude

• It shifts the graph horizontally

• It alters the speed at which the function completes a cycle

What is the period of the function y = -1/2 cos(3x)?

• 3π

• 2π/3

• π

• 2π

In the graph of y = -1/2 cos(3x), what does the amplitude represent?

• The total height of the graph

• The distance from the middle to the peak

• The width of one cycle

• The maximum height of the graph

How does the graph of y = -1/2 cos(3x) differ from a standard cosine graph?

• It is shifted vertically

• It has a different amplitude and period

• It is reflected over the x-axis

• It is shifted horizontally

What happens to the graph of y = -1/2 cos(3x) when x = 0?

• It starts at zero

• It starts at its minimum value

• It starts at its middle value

• It starts at its maximum value

How many cycles does y = -1/2 cos(3x) complete in the interval from 0 to 2π?

• One cycle

• Two cycles

• Four cycles

• Three cycles