In Exercises 67–68, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 4. y = cos πx + sin π/2 x

Verified step by step guidance

1. Identify the given function: \(y = \cos(\pi x) + \sin\left(\frac{\pi}{2} x\right)\), which is a sum of two trigonometric functions.
2. Choose values of \(x\) in the interval \(0 \leq x \leq 4\). For example, select integer values \(x = 0, 1, 2, 3, 4\) to start plotting points.
3. Calculate the \(y\)-coordinate for each chosen \(x\) by evaluating each trigonometric term separately: compute \(\cos(\pi x)\) and \(\sin\left(\frac{\pi}{2} x\right)\), then add these two results to get \(y\).
4. Plot each point \((x, y)\) on the coordinate plane using the values found in the previous step.
5. Connect the plotted points smoothly to graph the function over the interval \(0 \leq x \leq 4\), noting the periodic behavior of the cosine and sine components.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting points based on their values at specific x-coordinates. For functions like y = cos(πx) and y = sin(π/2 x), understanding their periodicity and amplitude helps in accurately sketching their curves over the given interval.

Sum of Functions (Adding y-coordinates)

When combining functions by adding their y-values, the resulting graph at each x is the sum of the individual function values. This method requires calculating y-values of each function separately at given x-points and then adding them to find the new y-coordinate for the combined function.

Trigonometric Function Periods and Frequencies

The period of a trigonometric function is the length of one complete cycle. For y = cos(πx), the period is 2, and for y = sin(π/2 x), the period is 4. Knowing these periods helps in determining key points and behavior of the functions within the interval 0 ≤ x ≤ 4.