Graph y=|cos(pix)|

Solution:

Step 1: Identify the Absolute Value Vertex

The vertex for the function $y = |\cos(\pi x)|$ is located at $(\frac{1}{2} + n, |\cos(\frac{\pi}{2} + \pi n)|)$.

Step 1.1: Determine the $x$-coordinate of the Vertex

Set the argument of the cosine function inside the absolute value to zero: $\cos(\pi x) = 0$.

Step 1.2: Solve for the $x$-coordinate

Find the value of $x$ by solving the equation $\cos(\pi x) = 0$.

Step 1.2.1: Use the Inverse Cosine Function

Apply the inverse cosine to both sides: $\pi x = \arccos(0)$.

Step 1.2.2: Simplify the Equation

Recognize that $\arccos(0) = \frac{\pi}{2}$, so $\pi x = \frac{\pi}{2}$.

Step 1.2.3: Isolate $x$

Divide both sides by $\pi$: $x = \frac{\frac{\pi}{2}}{\pi}$.

Step 1.2.4: Find the Second Solution

Since cosine is positive in the first and fourth quadrants, find the second solution by subtracting the reference angle from $2\pi$: $\pi x = 2\pi - \frac{\pi}{2}$.

Step 1.2.5: Simplify for $x$

Combine terms and divide by $\pi$ to get $x = \frac{3}{2}$.

Step 1.2.6: Determine the Period

Calculate the period of the cosine function: $T = \frac{2\pi}{|\pi|} = 2$.

Step 1.2.7: Apply the Period to Find All Solutions

The solutions repeat every $2$ units: $x = \frac{1}{2} + 2n$ and $x = \frac{3}{2} + 2n$ for any integer $n$.

Step 1.2.8: Combine Solutions

Express the general solution as $x = \frac{1}{2} + n$ for any integer $n$.

Step 1.3: Substitute $x$ into the Function

Replace $x$ in the original function with $\frac{1}{2} + n$: $y = |\cos(\pi(\frac{1}{2} + n))|$.

Step 1.4: Simplify the Function

Apply the cosine function to the expression: $y = |\cos(\frac{\pi}{2} + \pi n)|$.

Step 1.5: Finalize the Vertex Form

The absolute value vertex is $(\frac{1}{2} + n, |\cos(\frac{\pi}{2} + \pi n)|)$.

Step 2: Determine the Domain

The domain is all real numbers: Interval Notation: $(-\infty, \infty)$, Set-Builder Notation: $\{x | x \in \mathbb{R}\}$.

Step 3: Graph the Absolute Value Function

Use the vertices $(\frac{1}{2} + n, |\cos(\frac{\pi}{2} + \pi n)|)$ to plot the graph.

Knowledge Notes:

The problem involves graphing the function $y = |\cos(\pi x)|$. Here are the relevant knowledge points:

1. Absolute Value Function: The absolute value of a number is its distance from zero on the number line, without considering direction. The absolute value function graph is a V-shaped graph.

2. Cosine Function: The cosine function is a periodic function with a period of $2\pi$ radians. It oscillates between -1 and 1.

3. Transformations of Functions: The graph of $|\cos(\pi x)|$ is a transformation of the basic cosine function. The absolute value causes the graph to reflect across the x-axis wherever the cosine function would normally be negative.

4. Period of a Function: The period of a trigonometric function is the length of one complete cycle of the wave. For $y = \cos(bx)$, the period is given by $T = \frac{2\pi}{|b|}$.

5. Inverse Trigonometric Functions: These functions are used to find the angle that corresponds to a given trigonometric ratio. For example, $\arccos(0) = \frac{\pi}{2}$ radians.

6. Domain of a Function: The domain of a function is the set of all possible input values (x-values) for which the function is defined.

7. Graphing Techniques: To graph an absolute value function, it is often helpful to find the vertex and use symmetry, along with the period of the underlying trigonometric function, to plot additional points.

8. Set-Builder Notation: A notation to describe a set by stating the properties that its members must satisfy. For example, $\{x | x \in \mathbb{R}\}$ describes all real numbers.

9. Interval Notation: A notation to describe the set of numbers lying between two endpoints. The notation $(-\infty, \infty)$ represents all real numbers.

By understanding these concepts, one can graph the function $y = |\cos(\pi x)|$ and analyze its properties.