Problem 118 - Precalculus

Question

In Exercises \(115-118, A\) and \(B\) are positive real numbers. Find the \(x\) -intercepts of the function \(y=A \cos B x\)

Step-by-Step Solution

Answer

The \(x\)-intercepts are \(x = \frac{\pi}{2B} + \frac{n\pi}{B}\), where \(n\) is an integer.

Step 1: Understanding the Function

The function given is a cosine function: \(y = A \cos Bx\). Cosine functions have a repetitive wave pattern, and the intercepts can be found by setting the function equal to zero.

Step 2: Set Equation to Zero

To find the \(x\)-intercepts, set \(y=0\):\[A \cos Bx = 0\]This implies that \(\cos Bx = 0\).

Step 3: Solve Cosine Equation

The cosine function equals zero at \(\frac{\pi}{2} + n\pi\), where \(n\) is an integer. Therefore, set \(Bx = \frac{\pi}{2} + n\pi\).

Step 4: Solve for x

Solve the equation \(Bx = \frac{\pi}{2} + n\pi\) for \(x\): \[x = \frac{\frac{\pi}{2} + n\pi}{B}\]This gives the \(x\)-intercepts as \(x = \frac{\pi}{2B} + \frac{n\pi}{B}\), where \(n\) is any integer.

Key Concepts

Understanding X-Intercepts

The x-intercept of a function is the point where the graph of the function crosses the x-axis. At this point, the value of the function is zero, meaning the value of y is zero. Finding the x-intercepts involves solving for x when the function is equal to zero.
This is crucial in understanding and graphing a function, as x-intercepts are key components indicating changes or shifts in a graph's position. For the cosine function specifically, finding x-intercepts involves utilizing the unique periodic and symmetrical nature of trigonometric functions.

Introduction to Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are foundational in mathematics, particularly in geometry and wave theory. The cosine function, in particular, is associated with the shape of waves seen in sound and light. These functions are periodic and exhibit repeating patterns over specified intervals.
The basic form of the cosine function is \(y = \, ext{cos}(x)\). However, transformations like \(y = A \, ext{cos}(Bx)\) can stretch, compress, and shift the graph, depending on the values of A and B.

• A is the amplitude, dictating the height of the wave.
• B affects the period, determining how quickly the function repeats itself.

Understanding how these transformations affect the graph is key to mastering trigonometric functions.

Exploring Periodic Behavior

Periodic behavior refers to the repeating cycles seen in functions like cosine. The period is the distance over which the function's shape repeats. For the standard cosine function \(y = \, ext{cos}(x)\), the period is \(2\pi\). However, with transformations like \(y = A \, ext{cos}(Bx)\), the period alters to \(\frac{2\pi}{B}\).
This impacts how often the function crosses the x-axis or reaches a peak or valley. Periodic functions are crucial in describing oscillations, vibrations, and many natural phenomena.
When solving for x-intercepts, understanding the period helps predict where these intercepts occur along the x-axis, since they will reappear after each complete cycle.