Explain why when you multiply by a number greater than 1 inside the argument of a trigometric function, the period of the function decreases and when you multiply by a number less then one, the period increases
2x grows twice as fast as x, so the period is divided by 2
same for any k > 1
1/2 x grows half as fast, so the period doubles
When you multiply by a number greater than 1 inside the argument of a trigonometric function, such as sine or cosine, it affects the rate at which the function oscillates. The period of a trigonometric function is defined as the distance it takes to complete one full cycle.
Let's take the example of the sine function, but the same principle applies to other trigonometric functions as well. The general form of the sine function is given by:
y = sin(kx)
Here, k represents the multiplication factor inside the argument, and x represents the variable.
When k is greater than 1, it means that every x-value covers less distance to complete one full cycle of the sine function. This means that the period of the function decreases. Imagine stretching or compressing the graph horizontally, making each wave more narrow. As a result, the graph oscillates more frequently within the same interval.
Conversely, when k is less than 1, it means that every x-value covers more distance to complete one full cycle of the sine function. This leads to stretching the graph horizontally, making each wave wider. As a result, the graph oscillates less frequently within the same interval, which increases the period of the function.
In summary, multiplying by a number greater than 1 inside the argument of a trigonometric function decreases the period, while multiplying by a number less than 1 increases the period.
When you multiply the argument of a trigonometric function by a number greater than 1, such as sin(2x) or cos(3x), you are essentially compressing or squeezing the graph horizontally. To understand why this happens, let's take a look at the unit circle.
The unit circle is a circle with a radius of 1 and is used to define the values of trigonometric functions. The argument of a trigonometric function represents an angle, and multiplying it by a number affects the rate at which we move along the circle.
If we consider the sine function (sin(x)), for example, its graph represents the height of a point on the unit circle as we rotate around it. The period of the sine function is the distance required to complete one full cycle, which is 360 degrees or 2π radians.
Now, when we multiply the argument by a number greater than 1, say sin(2x), we are essentially doubling the speed at which we move around the unit circle. This means that the height of the sine function will complete one full cycle in half the distance, which results in a shorter period. Instead of taking 360 degrees or 2π radians to complete one cycle, it will now take only 180 degrees or π radians.
Similarly, when we multiply the argument by a number less than 1, let's say sin(0.5x), we are reducing the speed at which we move around the unit circle. This means that the height of the sine function will complete one full cycle in a longer distance, resulting in a longer period. Instead of taking 360 degrees or 2π radians, it will now take twice the amount, which is 720 degrees or 4π radians.
To summarize, multiplying the argument of a trigonometric function by a number greater than 1 compresses the graph horizontally and results in a shorter period, while multiplying by a number less than 1 stretches the graph horizontally and increases the period.