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40 Cards in this Set
- Front
- Back
a function is periodic if......
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its values repeat at regular intervals
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if f(x) is a periodic function with period of 2, then the graph of f(x+2) is.........
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identical to f(x)
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If a graph of f(x) is shifted horizontally by c units and the new graph is identical to the original graph, what is the period of f(x)?
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The period is c given that c is the smallest value for which f(x+c)=f(x)
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What is the midline of the function and what is a formula to calculate the midline of a function?
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The midline of a function is the horizontal line midway between the functions maximum and minimum values. Written as y=k where k is the midline.
A formula is: midline = (max + min)/2 |
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Define the amplitude and give a formula to calculate the amplitude of a function.
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the amplitude is the vertical distance between the functions's maximum value (or minimum value) and the midline.
A formula for the amplitude is: amplitude= |maximum - midline| or |minimum - midline| |
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If a graph starts at x = zero and begins increasing, then is the graph more easily modeled with a sin or cos formula?
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since sin(x) begins increasing immediately after x=0, then the graph is best modeled by a sin function.
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What are the standard formulas for a cos graph and a sin graph?
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y = Acos(B(x-h)) + k
y = Asin(B(x-h)) + k where A = amplitude B = 2pi/period h = horizontal shift with respect to the original function of sin(x) or cos(x). midline = the equation y = k period = 2pi/B |
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What is B in the formula
y=Asin(B(x-h))+k? |
B is the angular frequency; that is the number of cycles completed from 0<=x<=2pi
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True or false:
The midline is a distance while the amplitude is an equation in the form y=k |
False, the midline is not a distance, but rather a line y=k where k is the midpoint between the minimum and maximum values of the graph. The amplitude is a distance from the midline to the max or the min
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If y = cos(3t-4) then y is the graph of y = cos(t) shifted horizontally by how many units?
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Need to put equation in standard form:
y = cos(3t-4) = cos(3(t-4/3)) therefore h=4/3 so y is the graph of cos(t) shifted right by 4/3 units. |
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If a periodic function starts out at t=0 and begins decreasing immediately after t=0, is the function better modeled by a sin or cos function?
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a cos function since y=cos(t) begins decreasing immediately after t = 0
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If (x,y) is a point on the unit circle then what do x and y equal?
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x=cos(theta)
y=sin(theta) where theta is the angle that corresponds to the point (x,y) on the unit circle. |
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If (x,y) is a point on a circle with radius r, then what do x and y equal?
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x=rcos(theta)
y=rsin(theta) where theta is the angle that corresponds to the point (x,y) |
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If you are moving around the unit circle clockwise, will the angles you are sweeping out be negative or positive?
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Negative
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If you are moving around the unit circle counterclockwise will the angles you are sweeping out be negative or positive?
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positive
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If you are given x radians and asked to convert the x radians into degrees how would you do this?
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multiply the x radians by 180/pi
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If you are given x degrees and asked to convert these x degrees into radians how would you do this?
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multiply the x degrees by pi/180
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What is the arc length of a circle?
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It is the measure of the arc spanned by a given angle in the unit circle.
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What is the formula for the arc length?
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arc length =
(radius of the circle)(angle in radians) s = r(theta) where s=arc length r=radius of the circle theta=the corresponding angle in radians |
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What is the range of cos(x)?
What is the domain of cos(x)? |
Range: -1<=cos(x)<=1
Domain: all real numbers |
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What is the range of sin(x)?
What is the domain of sin(x)? |
Range: -1<=sin(x)<=1
Domain: all real numbers |
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What are the periods of both sin(x) and cos(x)?
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2pi radians or 360 degrees
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How many solutions of cos(t) = 0.4 does the inverse cos function give you?
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Only one solution
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How do you find all the solutions of
cos(t) = 0.4 for a given domain? |
Use the starting value you get from the cos inverse function, then find the other solution by symmetry. After that add and subtract the period times k where k=0,1,2,....
each k that you apply will give a different solution |
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since cos(input)=output then
inversecos(_______) = __________ |
output
input |
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find one solution to the equation using algebra:
2sin(x)=0.1 |
2sin(x)=0.1
sin(x)=0.1/2 inversesin(0.1/2) = x thus x = 0.05 |
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When trying to solve the equation:
sin(x)=0.8 you need to graph which two lines? |
Graph y = sin(x)
y = 0.8 Find where these graphs intersect and these will be the solutions to the equation. |
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Where are the inversecos and inversesin buttons located on your calculator?
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they are the functions above the cos and sin buttons and can be attained by pressing 2nd cos or 2nd sin respectively.
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What are other names for inversecos and inversesin?
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arccos and arcsin
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Is cos(x) even, odd, or neither?
What about sin(x)? |
cos(x) is even
sin(x) is odd |
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Is a parabola a periodic function?
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No, it does not repeat. It is a quadratic function.
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The angle of 2pi/3 radians is in which quadrant?
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2pi/3 radians
= (2pi/3)(180/pi) = 120 degrees Thus, 2pi/3 radians is in the 2nd quadrant |
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How is the period of cos(3x) related to the period of cos(x)?
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since B=3 for cos(3x) then
period = 2pi/B = 2pi/3 and the period of cos(x) is 2pi. Thus the period of cos(3x) is 1/3 that of the the period of cos(x) |
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Let sin(x) be transformed into
sin(0.5x) +2 List the transformations in the order in which they occurred |
1. period is increased by a factor of 1/0.5 = 2. Thus the period is doubled so the graph is horizontally stretched by a factor of 2 away from the y axis.
2. the graph is then shifted vertically up by two units |
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The function -3cos(-x) is the graph of cos(x) when it is...
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first flipped over the y axis
then vertically stretched by a factor of 3 then flipped over the x axis |
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What are the amplitude, period, midline of the function
y=-5sin(4t-9) + 7 |
amplitude=|-5| = 5
period = 2pi/B and B is found by (4t-9) = 4(t-9/4) so B = 4. Thus the period is pi/2 midline = 7 |
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what does arccos(cos(x)) equal?
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x
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what does cos(arccos(x)) equal?
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x
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what does sin(arcsin(x)) equal?
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x
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what does arcsin(sin(100x+2)) equal?
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100x + 2
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