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Graphing Sin/Cos Functions Guide (U2L1)
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Gravity
Key Concepts:
Terms in this set (51)
Amplitude
The distance between the midline and the maximum or minimum point (largest value attained, represented by the a in y=asinkx)
Midline
The horizontal line that divides the function symmetrically
Period
The distance required for a function to complete one full cycle
How to calculate amplitude
(Maximum)-(midline) OR (Absolute value of Maximum)-(midline)
Positive Cos Shape
Looks like c (for cos), note that a positive cos' vertex is less than the other points (like hole)
Negative Cos Shape
Looks like c (for cos), note that a negative cos' vertex is more than the other points (like hill)
Positive Sin Shape
Looks like s (for sin), note that a positive sin increases first to the maximum point and then decreases to the minimum point
Negative Sin Shape
Looks like s (for sin), note that a negative sin decreases first to the minimum point and then increases to the maximum point
What is the relationship between cos and sin?
Cos is 90 degrees (pi/2 radians) away from sin, meaning a sin function is simply a translation to the right of cos
How can one determine the minimum and maximum of a graph?
Add the amplitude to the midline for the maximum, subtract for the minimum
What is frequency? (include formula)
Frequency is represented by k in y=asin(kx). The formula is 2𝛑/k=Period. X varies between 0 and the period and frequency is what 2𝛑 is divided by to obtain said period
Equation for sin/cos functions
y=AMPsinFREQUENCY(x-phase shift)+MIDLINE or y=asink(x-b)+m
Is sint and cost periodic? Why?
Yes, they are, as sin(t+2n𝛑)=sint and cos(t+2n𝛑)=cost for any integer as n. The least such p being the period (for normal functions, is 2𝛑)
Periodic Function
A periodic function has no starting point and repeat infinitely in both directions with a specific pattern.
Review (What is 𝛑/3, 𝛑/6, and 𝛑/4?)
𝛑/3 is 60 degrees, 𝛑/6 is 30 degrees, 𝛑/4 is 45 degrees
Equation for the Midline
y=#
How to graph
Create a constant scale, label the axis, origin, equation, scale.
Graph the basic point first, such as the points at 0, the middle, and the full period (graph the minimum and maximum).
Make sure to take in account the positivity/negativity of a and the amplitude.
Next, if k is not 1, simply multiply k by an x coordinate and find the sin/cos of that new value. Multiply this by a. For instance, if y=2sin2x (period is 𝛑). ⅙ of 𝛑/2 is 𝛑/12. That multiplied by 2 is 𝛑/6 (30 degrees). Sin of 𝛑/6 is ½ and multiplied by 2, we get the y coordinate for ⅙ as 1.
Translations
Symbols outside the parenthesis/x represent up and down movements (up is +, down is -) f(x)-/+a, Symbols inside the parenthesis/directly operated on x represent right and left movements (- is right, + is left) f(x-/+a)
Reflections
Reflect over x-axis -f(x) (change y to opposite value, keep x the same) Reflect over y-axis f(-x) (change x to opposite value, keep y the same)
Dilations
Horizontal dilations: take the reciprocal for f(k*x) and multiply it by the x coordinate, Vertical dilations: take the k value in k*f(x) and multiply it by the y coordinate
What does it mean when sin/cos are negated?
Indicates a reflection over the x axis (x,y)-->(x,-y)
What does a fraction as an amplitude do to a function? A whole number, other than 1 or 0?
A whole number STRETCHES the function, while a fraction COMPRESSES the function (vertical dilation)
What does a fraction as a frequency do to a function? A whole number, other than 1 or 0?
A fraction expands the functions (stretches horizontally, cycle is slower), while a whole number compresses it (horizontally, cycle is faster)
Usual Methods of Graphing
USUALLY cos is 2 over 1 up and then 1 over 1 down. USUALLY sin is 1 over 1 up and then 1 over 1 down.
Phase Shift
y=asink(x-b) or y=asink(x+b). -b is to the right, while +b is to the left. However, when we write it individually a phase shift to the right is b, and to the left is -b.
Interval to graph one complete period is
[b, b+(2𝛑/k)]
# of blocks need to move (formula)
(12 blocks/Prd)=(x/phase shift)
What do you do when your equation looks like so: y=3/4cos(2x+2𝛑/3)
Factor out 2
√3
1.73 (𝛑/3)
√3/3
.58 (𝛑/6)
Tangent Function has Period...
𝛑 (as repeats every 𝛑), therefore period is equal to 𝛑/k
tan(x+_)=
𝛑, tanx
As tan gets closer to 𝛑/2, tanx (y value)...
Increase
As x--->𝛑/2...
tanx--->+infinity (meaning every real number can work that is smaller than pi/2)
As x--->-𝛑/2
tanx--->-infinity
Tanx
Increases without bound (never completely vertical, output it as high as we want it until 𝛑/2)
What is an asymptote?
Tan has an asymptote, which is a limit that tan cannot reach (has an equation, usually equal to ((2k+1)pi)/2
The amplitude and period for tan..
Has the same effect, simply stretch the function (without touching the asymptote)
A typical window for TAN is...
(-𝛑/2k, 𝛑/2k)
Differences between tan and cos/sin
1)No max or min 2)Period is 𝛑 3)Has asymptotes
If you have one section of sin/cos...
Multiply by 4 to get the period
How to graph tan
The period is the difference between two fractions. Therefore, if the period is 𝛑/2, then the tan function is initially graphed between 𝛑/4 and -𝛑/4. If the period is 𝛑/3, then the tan function is initially graphed between 𝛑/6 and -𝛑/6.
Note that the distance between these two x values is the period (that being 𝛑/2 or 𝛑/3). Then, usually we move over 1 and up .58 (𝛑/6) and then over 2 and up 1.73 (𝛑/3).
The half mark (𝛑/4) is usually where 1 is. If you have an amplitude other than 1 or 0, multiply it by these y values.
If you have a shift, simply move the function over that many (pertaining to units).
Midline is represented by
Axle Height
Amplitude is represented by
The radius (the diameter is measured from the maximum to the minimum point)
Frequency (Ferris)
Can be found by dividing 360 by the total time (make sure correct unit)
Period (Ferris)
Time (in specific unit)
Ferris Wheel problem will never have...
Negative points (Can't have negative time or negative distance). Will be represented by cos.
The cos function is NEGATIVE in a ferris wheel problem when...
The ferris wheel starts at the bottom (Because the wheel increases distance and then decreases, usually negative).
The cos function is POSITIVE in a ferris wheel problem when...
The ferris wheel starts at the top and thus decreases height and then increases
Regular Ferris Wheel Equation (start at bottom)
h=a-bcosc(t)
Characteristics of Tan
Period: pi Range: P Domain: P except [(2k+1)pi]/2
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