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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?
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?
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?
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?
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Which function represents a cosine function that is vertically translated downward?
Which function represents a cosine function that is vertically translated downward?
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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 $c > 0$?
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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 $y = ext{sin}(x) + d$?
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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 $a$?
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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?
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What is the first step in graphing sine and cosine functions?
What is the first step in graphing sine and cosine functions?
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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 $y = ext{cos}(x) + d$?
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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 $c$?
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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 $y = ext{cos}(x - c)$?
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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)?
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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?
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How is the period of the sine and cosine functions calculated?
How is the period of the sine and cosine functions calculated?
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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?
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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)?
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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?
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What characterizes the graphs of y = cos(bx) and y = cos(2bx)?
What characterizes the graphs of y = cos(bx) and y = cos(2bx)?
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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?
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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?
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What is the range of the sine and cosine functions?
What is the range of the sine and cosine functions?
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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?
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Which of the following statements is true regarding the sine function?
Which of the following statements is true regarding the sine function?
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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?
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What defines the amplitude of sine and cosine functions?
What defines the amplitude of sine and cosine functions?
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Which of the following behaviors is characteristic of the cosine function?
Which of the following behaviors is characteristic of the cosine function?
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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?
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What is the period of the sine and cosine functions?
What is the period of the sine and cosine functions?
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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?
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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?
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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 $y = 2 ext{sin } x$?
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What is the amplitude of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
What is the amplitude of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
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What is the period of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
What is the period of the function $y = -3 ext{cos } 6x - oldsymbol{ ext{π}} + 1$?
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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?
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What is the domain of the function $y = 2 ext{sin } 5x - 3oldsymbol{ ext{π}} - 7$?
What is the domain of the function $y = 2 ext{sin } 5x - 3oldsymbol{ ext{π}} - 7$?
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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?
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What is the range of the function $y = 3 ext{sin }(x - 0.3) + 2$?
What is the range of the function $y = 3 ext{sin }(x - 0.3) + 2$?
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Calculate the amplitude of the function $y = -3 ext{sin }(-7 + 5x)$.
Calculate the amplitude of the function $y = -3 ext{sin }(-7 + 5x)$.
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What is the period of the function $y = 3 ext{sin }(x - 0.3)$?
What is the period of the function $y = 3 ext{sin }(x - 0.3)$?
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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 $y = -2 ext{cos }(3x + 1oldsymbol{ ext{π}}) - 4$?
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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!
