In Exercises 14–15, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = sin x + cos 1/2 x

Guida verificata passo dopo passo

Identify the two separate functions involved: \( y_1 = \sin x \) and \( y_2 = \cos \frac{1}{2} x \). We will graph each function individually over the interval \( 0 \leq x \leq 2\pi \).

Create a table of values for \( y_1 = \sin x \) by choosing key points in the interval \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \) and calculating \( \sin x \) at these points.

Similarly, create a table of values for \( y_2 = \cos \frac{1}{2} x \) using the same \( x \)-values, but calculate \( \cos \left( \frac{1}{2} x \right) \) at each point.

Add the corresponding \( y \)-coordinates from the two tables to find the values of \( y = \sin x + \cos \frac{1}{2} x \) at each \( x \)-value. This means for each \( x \), compute \( y = y_1 + y_2 \).

Plot the points \( (x, y) \) obtained from the sums on the coordinate plane and connect them smoothly to graph the function \( y = \sin x + \cos \frac{1}{2} x \) over the interval \( 0 \leq x \leq 2\pi \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting points based on their values at various x-coordinates, typically over one or more periods. Understanding the shape and period of sine and cosine functions helps in accurately sketching their graphs.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Period and Frequency of Trigonometric Functions

The period of a trigonometric function is the length of one complete cycle. For y = cos(1/2 x), the period is 4π because the frequency is halved. Recognizing how coefficients inside the function affect the period is essential for correct graphing.

Video consigliato:

Percorso guidato

5:33

Period of Sine and Cosine Functions

Adding Functions by Summing y-Coordinates

When adding two functions, the resulting graph is found by adding their y-values at each x-coordinate. This method requires calculating y-values of each function separately and then summing them to get the combined function's graph.

Video consigliato:

Percorso guidato