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117 Cards in this Set
- Front
- Back
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A (larger/smaller) slope equates to a steeper line tilt.
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Larger
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A (positive/negative) slope equates to an upward tilt.
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Positive
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A (larger/smaller) slope equates to a shallower line tilt.
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Smaller
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A (positive/negative) slope equates to a downward tilt.
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Negative
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Given points (X1, Y1) and (X2, Y2) on a plane with a single line running through both of them, what is the equation to find the slope of the line.
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m = (y2 - y1) / (x2 - x1)
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What is the equation of a line in slope-intercept form?
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y = mx + b
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In the line equation y = mx + b, what does m represent?
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Slope
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In the line equation y = mx + b, what does b represent?
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Y-intercept
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What is the equation of a line in point-slope form?
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y - y1 = m(x - x1)
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Two lines are parallel if their slopes are ____ or both are ____.
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Equal, vertical
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Two lines are perpendicular if their slopes are ____.
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Negative reciprocals.
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A function is a special relation where ____ element(s) of Set A is/are paired with ____ element(s) of Set B.
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Every, exactly one
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Functions containing no variables.
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Constant functions
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The domain of a function from Set A to Set B is ____.
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All of Set A
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The range of a function from Set A to Set B is ____.
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All or part of Set B
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A graph must pass the ____ to be considered a function.
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Virtual line test
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What is the virtual line test?
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Any vertical line can touch the graph either once or not at all.
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A graph's horizontal extension shows the function's (domain/range).
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Domain
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A graph's vertical extension shows the function's (domain/range).
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Range
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When plotting points on a graph, (open/closed) dots should be used for values that include the dot.
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Closed
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When plotting points on a graph, (open/closed) dots should be used for values that do not include the dot.
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Open
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A graph has x-axis symmetry if ____.
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f(x) = 0
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A graph has y-axis symmetry if ____.
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f(x) = f(-x)
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A graph has symmetry with respect to origin if ____.
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f(x) = -f(x)
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a(-x)^y
If y is an even power this is equivalent to ____. |
a(x)^y
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a(-x)^y
If y is an odd power this is equivalent to ____. |
-ax^y
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f + g(x) can be rewritten as ____.
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f(x) + g(x)
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f - g(x) can be rewritten as ____.
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f(x) - g(x)
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fg(x) can be rewritten as ____.
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f(x)g(x)
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(f/g)(x) can be rewritten as ___.
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f(x) / g(x)
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f o g(x) can be rewritten as ____.
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f( g(x) )
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A function has an inverse if it ____.
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Passes the horizontal line test
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For inverses f(x) and f^-1 (x), if f(x) contains the point (a,b) then f^-1 (x) will contain the point ____.
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(b,a)
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How do you find the inverse of f(x)?
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Replace f(x) with x, replace x with y, and solve for y. Replace y with f^-1 (x).
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Function f(x) has identity i(x) if ____.
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f o i(x) = f(x)
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A quadratic function has a minimum function if the graph opens ____.
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Up
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A quadratic function has a maximum function if the graph opens ____.
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Down
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On a graph of a polynomial function, the leading coefficient is positive and the polynomial degree is even. How will the graph behave?
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Both ends will point up
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On a graph of a polynomial function, the leading coefficient is positive and the polynomial degree is odd. How will the graph behave?
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The graph will go down and then up.
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On a graph of a polynomial function, the leading coefficient is negative and the polynomial degree is even. How will the graph behave?
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Both ends will point down.
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On a graph of a polynomial function, the leading coefficient is negative and the polynomial degree is odd. How will the graph behave?
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The graph will go up and then down.
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The Rational Zero Theorem states that if p/q is a zero, then p is a divisor of the ____ and q is a divisor of the ____.
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Constant term, leading coefficient
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Descartes' Rule of Signs states that the maximum positive zeros in f(x) is ____.
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The number of sign changes in f(x)
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Descartes' Rule of Signs states that the possible positive zeros in f(x) is ____.
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The maximum positive zeros minus an even whole number.
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Descartes' Rule of Signs states that the maximum negative zeros in f(x) is ____.
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The number of sign changes in f(-x).
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Descartes' Rule of Signs states that the possible negative zeros in f(x) is ____.
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The maximum negative zeros minus an even whole number.
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The Upper and Lower Bounds Theorem states that the results of doing synthetic division can determine if the divisor is a bound.
How can you determine if the divisor is an upper bound? |
The bottom row of the division will all be non-negative.
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The Upper and Lower Bounds Theorem states that the results of doing synthetic division can determine if the divisor is a bound.
How can you determine if the divisor is a lower bound? |
The bottom row of the division will alternate between non-positive and non-negative.
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How many y-intercepts does every polynomial function have?
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One
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What is a rational function?
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A function that can be written as one polynomial divided by another.
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What is a vertical asymptote?
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A break in the graph of a rational function where the denominator equals 0.
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How do you find the y-intercept of a rational function?
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By setting x to 0.
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Where is the horizontal asymptote in the graph of a rational function where the degree of the numerator is larger than the degree of the denominator?
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There is no horizontal asymptote.
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Where is the horizontal asymptote in the graph of a rational function where the degree of the numerator is smaller than the degree of the denominator?
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y = 0
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Where is the horizontal asymptote in the graph of a rational function where the degree of the numerator equals the degree of the denominator?
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y = a_n / b_m
a_n = leading coefficient of the numerator a_m = leading coefficient of the denominator |
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How do you find the x-intercepts of a rational function?
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By setting the numerator to 0.
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What is the formula for calculating compound interest? Define all variables.
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A = P(1 + r/n)^nt
P = Dollars invested r = Yearly rate n = Periods per year t = Time in years |
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What is the formula for continuous compounding? Define all variables.
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A = Pe^(rt)
P = Dollars invested e = Euler's number r = Yearly rate t = Time in years |
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What is the formula for general growth? Define all variables.
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n(t) = n_0 e^(rt)
n = compounding periods t = time n_0 = beginning amount r = rate |
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What is the formula for general decline? Define all variables.
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n(t) = n_0 e^(-rt)
n = compounding periods t = time n_0 = beginning amount r = rate |
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Rewrite log_a x = y in exponential form.
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a^y = x
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Rewrite a^y = x in logarithmic form.
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log_a x = y
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Rewrite log_a a^x.
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x
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Rewrite a^(log_a x).
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x
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Rewrite the nth root of a^m?
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a^(m/n)
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Rewrite a^(m/n).
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nth root of a^m
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Rewrite 1/(a^m).
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a^(-m)
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Rewrite a^(-m).
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1/(a^m)
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Rewrite log_3 (x+1) = 4 in exponential form.
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3^4 = x+1
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If there are logs in both sides of an equation, when do they cancel?
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When their bases are the same.
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In log_x a, what must a be?
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a > 0
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Rewrite log_b mn.
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log_b m + log_b n
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Rewrite log_b m + log_b n.
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Log_b mn
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Rewrite log_b (m/n).
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log_b m - log_b n
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Rewrite log_b m - log_b n.
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log_b (m/n)
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Rewrite log_b m^t.
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t log_b m
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Rewrite t log_b m.
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log_b m^t
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What is the change of base formula for logarithm log_b x? Let a equal the new base.
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log_b x = (log_a x) / (log_a b)
b = old base a = new base |
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How do you solve an equation when exponents with variables are on both sides?
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1. Take log or ln of each side.
2. Use log property three to remove exponents from logs. 3. Move x to one side. 4. Factor x. 5. Divide to isolate x. |
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What are two ways to solve systems of equations?
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Substitution and elimination by addition.
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If two systems of equations do not yield an intersection, what might they be? How do you determine which is correct?
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Parallel or the same line. If the result is true, it is the same line. If the result is false, they are parallel.
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What is the memory aid for the right-angle trigonometry function?
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SOHCAHTOA
Some Old Helmets Can Allow Head Trauma On Accidents Sin = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent |
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Reproduce the unit circle memory aid for sin, cos, tan, sec, and csc.
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tan is sin/cos
sec is 1/cos cot is cos/sin csc is 1/sin |
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Where is the unit circle centered?
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The origin
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What is the radius of the unit circle?
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1
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What are the two sides of an angle?
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Initial side and terminal side
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Which direction do positive angles rotate?
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Counter-clockwise
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Which direction do negative angles rotate?
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Clockwise
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What are radians based on?
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The circle's circumference
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What is the circumference of a unit circle?
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2pi
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How many radians is a 360 degree circle?
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2pi radians
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How many radians is a 180 degree circle?
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pi radians
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What are the three instances in which an angle can be co-terminal?
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Terminal sides are identical
Difference between angles is a multiple of 360 degrees Difference between angles is a multiple of 2pi |
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How many radians are in one degree?
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pi/180
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How many degrees are in one radian?
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180/pi
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What does theta's reference angle represent?
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The smallest angle between the terminal side of the angle and the x-axis
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How are the quadrants numbered?
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II I
III IV |
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For angle theta, the point where the terminal side intersects the circumference of the unit circle, how do you determine (x,y)?
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x = cos theta
y = sin theta |
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What is the equation of the unit circle?
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x^2 + y^2 = 1
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In which quadrants of the unit circle is cosine positive?
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I and IV
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In which quadrants of the unit circle is cosine negative?
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II and III
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In which quadrants of the unit circle is sine positive?
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I and II
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In which quadrants of the unit circle is sine negative?
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III and IV
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For what range of x is y positive on the graph of f(x) = sin x?
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0 through pi
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For what range of x is f(x) negative on the graph of f(x) = sin x?
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pi through 2pi
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For what range of x is f(x) positive on the graph of f(x) = cos x?
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3pi/2 through 2pi + pi/2
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For what range of x is y negative on the graph of f(x) = cos x?
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pi/2 through 3pi/2
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What translation occurs between f(x) = cos x and f(x) = sin x?
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The graph is shifted pi/2 units horizontally
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How are f(x) = c + sin x and f(x) = c + cos x translated in relation to c?
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They are shifted up c units
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How are f(x) = c sin x and f(x) = c cos x translated in relation to c?
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They are stretched by a factor of c
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How are f(x) = sin (x - c) and f(x) = cos (x - c) translated in relation to c?
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They are shifted horizontally by c units
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In a sine or cosine transformation, the degree of vertical stretching or compressing.
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Amplitude
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In a sine or cosine transformation, the horizontal shift.
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Phase shift
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Write the identities of the right-angle trigonometric functions.
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sin = O/H
cos = A/H tan = O/A cot = A/O sec = H/A csc = H/O |
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When can trigonometric functions have inverses?
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When their domain is limited
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What are two names for the inverse of sin?
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arcsin and sin^-1
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Rewrite arcsin theta = x to find the value of theta.
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sin x = theta
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