Question

If \(y=f(x)\) has a period of \(6,\) determine the period of \(y=f\left(\frac{1}{2} x\right)\)

Short answer

The period is 12.

Step-by-step solution

Understand the given period

The function given is \( y = f(x) \) with a period of \( 6 \). This means that for any value of \( x \), \( f(x) = f(x + 6) \).

Substitute in the new function

We need to determine the period of the new function \( y = f\left(\frac{1}{2} x\right) \). Let's consider when \( f(x) \) has the same period as \( f\left(\frac{1}{2} x\right) \) by substituting \( \frac{1}{2} x \).

Set up the equation for the new period

To find the period \( T \) of \( f\left(\frac{1}{2} x\right) \), we need to find \( T \) such that \( f\left(\frac{1}{2} x + \frac{1}{2} T\right) = f\left(\frac{1}{2} x\right) \). This implies the argument of the function must change by 6 units due to the original period: \[ \frac{1}{2} T = 6 \]

Solve for the new period

Now solve \( \frac{1}{2} T = 6 \) for \( T \): \[ T = 6 \times 2 = 12 \] Hence, the period of the function \( y = f\left(\frac{1}{2} x\right) \) is \( 12 \).

Key concepts

function period

When working with periodic functions, understanding the period is crucial. The period of a function is the interval over which the function's values repeat. For example, if a function has a period of 6, this means that it repeats its values every 6 units. Mathematically, for a function \(f(x)\) with period \(P\), we have \(f(x) = f(x + P)\) for any value of \(x\).

Knowing the period helps us predict the behavior of the function over larger intervals. It's especially important in problems involving function transformations, where the period can change. Recognizing these changes allows us to grasp the new pattern of repetition.

periodic functions

A periodic function is one that repeats its values at regular intervals, known as periods. Examples include trigonometric functions like sine and cosine, which have periods of \(2\pi\). Understanding periodic functions is essential in various fields such as signal processing, physics, and engineering.

To determine the period of a transformed function, we need to understand how the transformation affects the original period. For instance, if we have a function \(f(x)\) with period 6, and we transform it by scaling the input, such as \(f(\frac{1}{2} x)\), the period changes accordingly. The transformation can stretch or shrink the period depending on the scaling factor. In general, if the transformation is of the form \(f(ax)\), then the new period \(P_{\text{new}}\) can be found using the relationship \(P_{\text{new}} = P_{\text{original}} \times |a|\).

function transformations

Function transformations involve shifting, stretching, or compressing the graph of a function. In the exercise, we looked at the transformation \(y = f(\frac{1}{2} x)\). This transformation scales the input \(x\) by \(\frac{1}{2}\), effectively stretching the graph horizontally.

To find the new period of the transformed function, we set up an equation based on the changed input. We know the original period is 6, so we use the equation \(\frac{1}{2} T = 6\) to find the new period \(T\). Solving this equation gives us \(T = 12\). Therefore, the period of \(y = f(\frac{1}{2} x)\) is 12.

Understanding these transformations is vital for graphing functions correctly and predicting their behavior after transformations. It's a key skill across mathematics and its applications in real-world problems.