University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

358

Pages:

6
Author:

Bruno Cain

Graphing Sine and Cosine Functions For use with Exploration 9.4 Essential Question What are the characteristics of the graphs of the sine and cosine functions? 1 EXPLORATION: Graphing the Sine Function Go to BigIdeasMath.com for an interactive tool to investigate this exploration. Work with a partner. a. Complete the table for (y = \sin x) where x is an angle measure in radians. x y = sin x -2π 0 -7π/4 √2/2 -3π/2 1 -5π/4 √2/2 -π 0 -3π/4 -√2/2 -π/2 -1 -π/4 -√2/2 0 0 π/4 √2/2 π/2 1 3π/4 √2/2 π 0 5π/4 -√2/2 3π/2 -1 7π/4 -√2/2 2π 0 b. Plot the points (x, y) from part (a). Draw a smooth curve through the points to sketch the graph of (y = \sin x). graph of y = sin x c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over -2π ≤ x ≤ 2π. x-intercepts: -2π, -π, 0, π, 2π local max: π/2, 5π/2 local min: -π/2, 3π/2 Increasing: -3π/2 < x < π/2, 3π/2 < x < 5π/2 Decreasing: -2π < x < -3π/2, π/2 < x < 3π/2 2 EXPLORATION: Graphing the Cosine Function a. Complete the table for (y = \cos x) using the same values of x as those used in Exploration 1. x y = cos x -2π 1 -7π/4 √2/2 -3π/2 0 -5π/4 -√2/2 -π -1 -3π/4 -√2/2 -π/2 0 -π/4 √2/2 0 1 π/4 √2/2 π/2 0 3π/4 -√2/2 π -1 5π/4 -√2/2 3π/2 0 7π/4 √2/2 2π 1 b. Plot the points (x, y) from part (a) and sketch the graph of (y = \cos x) graph of y = cos x c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals 1 for which the function is increasing or decreasing 2 over -2π ≤ x ≤ 2π. ebin.pub ebin.pub x-intercepts: -3π/2, -π/2, π/2, 3π/2 local max: 0, 2π local min: -π, π Increasing: -π < x < 0, π < x < 2π Decreasing: -2π < x < -π, 0 < x < π Communicate Your Answer What are the characteristics of the graphs of the sine and cosine functions? min/max are the same repeats pattern Describe the end behavior of the graph of (y = \sin x). oscillates b/w 1 and -1 Notetaking with Vocabulary For use after Lesson 9.4 In your own words, write the meaning of each vocabulary term. amplitude high or low periodic function regularly cycle how long does it take to repent period time frame phase shift transformation midline middle of Core Concepts Characteristics of (y = \cos x) and (y = \sin x) The domain of each function is all real numbers. The range of each function is -1 ≤ y ≤ 1. So, the minimum value = -1 and the maximum value = 1. The amplitude of the graph of each function is one-half of the difference of the maximum value and the minimum value, or [\frac{1}{2}[1-(-1)]=1.] how far up or down do you go from the middle Each function is periodic, which means that its graph has a repeating pattern. The shortest repeating portion of the graph is called a cycle. The horizontal length of each cycle is called the period. The graph of each function has a period of 2π. The x-intercepts for (y = \sin x) occur when (x = 0, ±π, ±2π, ±3π, ...) The x-intercepts for (y = \cos x) occur when (x = ±\frac{\pi}{2}, ±\frac{3\pi}{2}, ±\frac{5\pi}{2}, ±\frac{7\pi}{2}, ...)