Graphs of Sine and Cosine Functions

Questions and Answers

What factor determines the vertical stretching or shrinking of the graph of a sine or cosine function?

The period (b)

The vertical shift (d)

The phase shift (c)

The amplitude (a) (correct)

What is the formula for the period of a sine or cosine function?

$rac{2 ext{π}}{b}$ (correct)

$rac{b}{2 ext{π}}$

$ ext{π}|b|$

$2 ext{π}|b|$

What does the phase shift (c) in the function y = a sin(b(x - c)) + d indicate?

Changes in the period of the graph

A horizontal shift of the graph (correct)

A vertical translation of the graph

The amplitude adjustments

What is the range of the function y = a sin(b(x - c)) + d?

$-|a| + d$ to $|a| + d$ Signup and view all the answers

Which function represents a cosine function that is vertically translated downward?

y = cos(x) - 3 Signup and view all the answers

What happens to the graph of $y = ext{sin}(x - c)$ when $c > 0$?

The graph shifts $c$ units right. Signup and view all the answers

What is the effect of a vertical shift determined by $d$ in the graph of $y = ext{sin}(x) + d$?

If $d < 0$, the graph is shifted downwards. Signup and view all the answers

In graphing sine and cosine functions, what is the role of the parameter $a$?

It specifies the amplitude of the graph. Signup and view all the answers

How is the range of the function $y = ext{sin}(x) + d$ determined?

It is $[-a + d, a + d]$. Signup and view all the answers

What is the first step in graphing sine and cosine functions?

Construct a table of values. Signup and view all the answers

Which of the following describes the effect of a negative value of $d$ in $y = ext{cos}(x) + d$?

The graph shifts downwards. Signup and view all the answers

When examining the graphs of $y = ext{sin}(x + c)$ and $y = ext{sin}(x - c)$, what is the effect of changing the sign of $c$?

The graph shifts in opposite horizontal directions. Signup and view all the answers

What common mistake might students make regarding the phase shift in $y = ext{cos}(x - c)$?

Assuming it shifts left when $c$ is positive. Signup and view all the answers

What is the effect of increasing the value of b in the functions y = sin(bx) and y = cos(bx)?

The period of the graphs decreases and the graphs shrink horizontally. Signup and view all the answers

What does the phase shift in the functions y = sin(x - c) and y = cos(x - c) indicate?

It shows how the graph is shifted horizontally from the base function. Signup and view all the answers

How is the period of the sine and cosine functions calculated?

The period is calculated as $\frac{2\pi}{b}$. Signup and view all the answers

What happens to the graphs of y = sin(bx) and y = cos(bx) when 0 < b < 1?

The graphs stretch horizontally. Signup and view all the answers

What is true about the graphs of y = sin(-bx) and y = -sin(bx)?

They represent the same function. Signup and view all the answers

When comparing the graphs of y = sin(x) and y = sin(x) + $\frac{\pi}{2}$, what change occurs?

The graph shifts upwards. Signup and view all the answers

What characterizes the graphs of y = cos(bx) and y = cos(2bx)?

y = cos(2bx) has a shorter period than y = cos(bx). Signup and view all the answers

If c in y = sin(x - c) is negative, what does this indicate?

The graph is shifted to the left. Signup and view all the answers

What do the graphs of y = sin(bx) and y = cos(bx) have in common?

They are both periodic functions with the same amplitude. Signup and view all the answers

What is the range of the sine and cosine functions?

−1 ≤ y ≤ 1 Signup and view all the answers

What happens to the graph of y = a sin x when a > 1?

The graph stretches vertically. Signup and view all the answers

Which of the following statements is true regarding the sine function?

It is an odd function. Signup and view all the answers

How does the graph of y = −a sin x compare to the graph of y = a sin x?

They are vertical reflections of each other. Signup and view all the answers

What defines the amplitude of sine and cosine functions?

It is defined as |a| in the functions y = a sin x and y = a cos x. Signup and view all the answers

Which of the following behaviors is characteristic of the cosine function?

It intersects the y-axis at its maximum value. Signup and view all the answers

If the value of a in y = a cos x is negative, what characteristic will the graph exhibit?

It will invert the graph. Signup and view all the answers

What is the period of the sine and cosine functions?

2π Signup and view all the answers

What effect does increasing the value of a in y = a sin x have on the vertical distance from the x-axis?

It increases the distance. Signup and view all the answers

What is true about the function y = a sin x when a < 1?

The graph shrinks vertically. Signup and view all the answers

What is the equation of the sine function shifted $2oldsymbol{ ext{π}}$ units to the right and $5$ units downward that has the same shape as $y = 2 ext{sin } x$?

$y = 2 ext{sin }(x - 2oldsymbol{ ext{π}}) - 5$ Signup and view all the answers

What is the amplitude of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?

$3$ Signup and view all the answers

What is the period of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?

$rac{2oldsymbol{ ext{π}}}{3}$ Signup and view all the answers

For the function $y = -2 ext{sin } x + 0.4$, what is the range of the function?

$[-2.4, 2.4]$ Signup and view all the answers

What is the domain of the function $y = 2 ext{sin } 5x - 3oldsymbol{ ext{π}} - 7$?

All real numbers Signup and view all the answers

What would be the equation of the sine function with the same shape as $y = - ext{sin } 3x$ that is shifted $oldsymbol{ ext{π}}$ units to the left and $8$ units upward?

$y = - ext{sin }(3x + oldsymbol{ ext{π}}) + 8$ Signup and view all the answers

What is the range of the function $y = 3 ext{sin }(x - 0.3) + 2$?

$[1, 5]$ Signup and view all the answers

Calculate the amplitude of the function $y = -3 ext{sin }(-7 + 5x)$.

$3$ Signup and view all the answers

What is the period of the function $y = 3 ext{sin }(x - 0.3)$?

$2oldsymbol{ ext{π}}$ Signup and view all the answers

What is the transformed range of the function $y = -2 ext{cos }(3x + 1oldsymbol{ ext{π}}) - 4$?

$[-6, -2]$ Signup and view all the answers

Study Notes

Sine and Cosine Functions Overview

Sine and cosine functions are defined for all real numbers.
Sine function notation: 𝑦 = sin 𝑥.
Cosine function notation: 𝑦 = cos 𝑥.

Graph Characteristics

Domain: All real numbers (ℝ).
Range: -1 ≤ 𝑦 ≤ 1.
Period: Each function completes one cycle every 2𝜋.

Amplitude and Multiple Functions

Amplitude (|𝑎|) determines the vertical distance from the x-axis to the highest/lowest point.
If |𝑎| < 1, functions shrink vertically; if |𝑎| > 1, functions stretch vertically.
Negative 𝑎 results in a vertical reflection across the x-axis.

Function Types

Odd Function: 𝑓(−𝑥) = −𝑓(𝑥), applies to the sine function.
Even Function: 𝑓(−𝑥) = 𝑓(𝑥), applies to the cosine function.

Period Determination

Period determined by the value of 𝑏 in the equations 𝑦 = sin(𝑏𝑥) and 𝑦 = cos(𝑏𝑥).
Period formula: 2𝜋/|𝑏|.
If 𝑏 > 1, the graph shrinks horizontally; if 0 < 𝑏 < 1, it stretches horizontally.

Phase Shift

Phase shift indicated by 𝑐 in functions 𝑦 = sin(𝑥 − 𝑐) and 𝑦 = cos(𝑥 − 𝑐).
If 𝑐 < 0, the graph shifts left; if 𝑐 > 0, the graph shifts right.

Vertical Shift

Determined by 𝑑 in functions 𝑦 = sin 𝑥 + 𝑑 and 𝑦 = cos 𝑥 + 𝑑.
If 𝑑 < 0, the graph shifts down; if 𝑑 > 0, it shifts up.
New range: −|𝑎| + 𝑑 ≤ 𝑦 ≤ |𝑎| + 𝑑.

Steps for Graphing

Construct a table of values to define the function.
Plot points on a coordinate plane; create a smooth curve.
Extend the graph horizontally by repeating cycles.

Practical Application

Example of function transformation: For the function 𝑦 = 𝟐 sin 𝑥 shifted by 2𝜋 right and 5 down, the resulting equation is 𝑦 = ±𝟐 sin(𝑥 − 2𝜋) − 𝟓.
Determine amplitude, period, domain, and range for various functions, e.g., 𝑦 = −𝟑 cos(𝟔𝑥 − 𝜋 + 𝟏).

Summary of Key Properties

Periodic functions repeat every 2𝜋/|𝑏|.
Amplitude controls vertical transformation.
Phase shift affects horizontal transformation.
Vertical shift modifies the graph's placement on the y-axis.
Domain includes all real numbers; range defined as −|𝑎| + 𝑑 ≤ 𝑦 ≤ |𝑎| + 𝑑.

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Description

This quiz explores the graphs of the sine and cosine functions, y = sin x and y = cos x. It covers their characteristics, shapes, and key properties. Test your understanding of these fundamental mathematical concepts!