Podcast
Questions and Answers
What factor determines the vertical stretching or shrinking of the graph of a sine or cosine function?
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?
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?
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?
What is the range of the function y = a sin(b(x - c)) + d?
Which function represents a cosine function that is vertically translated downward?
Which function represents a cosine function that is vertically translated downward?
What happens to the graph of $y = ext{sin}(x - c)$ when $c > 0$?
What happens to the graph of $y = ext{sin}(x - c)$ when $c > 0$?
What is the effect of a vertical shift determined by $d$ in the graph of $y = ext{sin}(x) + d$?
What is the effect of a vertical shift determined by $d$ in the graph of $y = ext{sin}(x) + d$?
In graphing sine and cosine functions, what is the role of the parameter $a$?
In graphing sine and cosine functions, what is the role of the parameter $a$?
How is the range of the function $y = ext{sin}(x) + d$ determined?
How is the range of the function $y = ext{sin}(x) + d$ determined?
What is the first step in graphing sine and cosine functions?
What is the first step in graphing sine and cosine functions?
Which of the following describes the effect of a negative value of $d$ in $y = ext{cos}(x) + d$?
Which of the following describes the effect of a negative value of $d$ in $y = ext{cos}(x) + d$?
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 $c$?
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 $c$?
What common mistake might students make regarding the phase shift in $y = ext{cos}(x - c)$?
What common mistake might students make regarding the phase shift in $y = ext{cos}(x - c)$?
What is the effect of increasing the value of b in the functions y = sin(bx) and y = cos(bx)?
What is the effect of increasing the value of b in the functions y = sin(bx) and y = cos(bx)?
What does the phase shift in the functions y = sin(x - c) and y = cos(x - c) indicate?
What does the phase shift in the functions y = sin(x - c) and y = cos(x - c) indicate?
How is the period of the sine and cosine functions calculated?
How is the period of the sine and cosine functions calculated?
What happens to the graphs of y = sin(bx) and y = cos(bx) when 0 < b < 1?
What happens to the graphs of y = sin(bx) and y = cos(bx) when 0 < b < 1?
What is true about the graphs of y = sin(-bx) and y = -sin(bx)?
What is true about the graphs of y = sin(-bx) and y = -sin(bx)?
When comparing the graphs of y = sin(x) and y = sin(x) + $\frac{\pi}{2}$, what change occurs?
When comparing the graphs of y = sin(x) and y = sin(x) + $\frac{\pi}{2}$, what change occurs?
What characterizes the graphs of y = cos(bx) and y = cos(2bx)?
What characterizes the graphs of y = cos(bx) and y = cos(2bx)?
If c in y = sin(x - c) is negative, what does this indicate?
If c in y = sin(x - c) is negative, what does this indicate?
What do the graphs of y = sin(bx) and y = cos(bx) have in common?
What do the graphs of y = sin(bx) and y = cos(bx) have in common?
What is the range of the sine and cosine functions?
What is the range of the sine and cosine functions?
What happens to the graph of y = a sin x when a > 1?
What happens to the graph of y = a sin x when a > 1?
Which of the following statements is true regarding the sine function?
Which of the following statements is true regarding the sine function?
How does the graph of y = −a sin x compare to the graph of y = a sin x?
How does the graph of y = −a sin x compare to the graph of y = a sin x?
What defines the amplitude of sine and cosine functions?
What defines the amplitude of sine and cosine functions?
Which of the following behaviors is characteristic of the cosine function?
Which of the following behaviors is characteristic of the cosine function?
If the value of a in y = a cos x is negative, what characteristic will the graph exhibit?
If the value of a in y = a cos x is negative, what characteristic will the graph exhibit?
What is the period of the sine and cosine functions?
What is the period of the sine and cosine functions?
What effect does increasing the value of a in y = a sin x have on the vertical distance from the x-axis?
What effect does increasing the value of a in y = a sin x have on the vertical distance from the x-axis?
What is true about the function y = a sin x when a < 1?
What is true about the function y = a sin x when a < 1?
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 $y = 2 ext{sin } x$?
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 $y = 2 ext{sin } x$?
What is the amplitude of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
What is the amplitude of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
What is the period of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
What is the period of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
For the function $y = -2 ext{sin } x + 0.4$, what is the range of the function?
For the function $y = -2 ext{sin } x + 0.4$, what is the range of the function?
What is the domain of the function $y = 2 ext{sin } 5x - 3oldsymbol{ ext{π}} - 7$?
What is the domain of the function $y = 2 ext{sin } 5x - 3oldsymbol{ ext{π}} - 7$?
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?
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?
What is the range of the function $y = 3 ext{sin }(x - 0.3) + 2$?
What is the range of the function $y = 3 ext{sin }(x - 0.3) + 2$?
Calculate the amplitude of the function $y = -3 ext{sin }(-7 + 5x)$.
Calculate the amplitude of the function $y = -3 ext{sin }(-7 + 5x)$.
What is the period of the function $y = 3 ext{sin }(x - 0.3)$?
What is the period of the function $y = 3 ext{sin }(x - 0.3)$?
What is the transformed range of the function $y = -2 ext{cos }(3x + 1oldsymbol{ ext{π}}) - 4$?
What is the transformed range of the function $y = -2 ext{cos }(3x + 1oldsymbol{ ext{π}}) - 4$?
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!
