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90 Cards in this Set
- Front
- Back
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A function is:
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a rule that assigns each element in one set to a unique element in a second set.
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A function is a set of:
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ordered pairs in which no two ordered pairs have the same first coordinate and different second coordinate.
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x is the ____ variable:
y is the ____ variable: |
x is the independent variable, y is the dependent variable.
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A set of ordered pairs is called:
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a relation
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What is the vertical line test?
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A test to see if a graph is a function. If a vertical line crosses a graph more than once, then the graph is not the graph of a function.
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The Vertical line test: a graph is the graph of a function IFF:
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there is no vertical line that crosses the graph more than once.
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Every ____ line is the graph of a function.
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every non-vertical line.
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The domain is:
The range is: |
The domain is the set of all the x values of the ordered pairs of a relation.
The range is the set of all the y values of the ordered pairs of a relation. |
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f = {(2,5)} so: f( ) =
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so: f(2) = 5
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The Phrase "is a function of" means:
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"is determined by." i.e.. y is a function of x means: the value of y is determined by the value of x. or "Can y be uniquely determined from x?"
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must x and y of a function be real numbers?
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yes
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What is the avg. rate of change formula?
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∆y / ∆x ie.. the slope formula.
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The Difference Quotient is:
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the expression:
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How do you determine if an equation defines "y as a function of x?"
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Solve each equation for y in terms of x. If two or more values of y can be obtained for a given x, the equation is not a function.
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How do you determine the domain and range of a relation?
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Change x or y into an inequality as needed.
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The average rate of change between two points on a graph is called:
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the difference quotient
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Anytime there is a line with a positive or negative slope, the Domian and Range are:
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the same
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What is the equation of the Square function? Graph?
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y = x² or y = -x²
The graph is a parabola opening either up or down. |
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What is the equation of the Square-root function? Graph?
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y = √x or y = -√x
The graph is a half-parabola opening to the right either above or below the x-axis. |
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What is the equation of a parabola opening to the right?
Is there an equation for a parabola opening to the left? |
x = y²
Yes, x = -y² |
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What is the equation of the Cube function? Graph?
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y = x³ or y = -x³
The graph is a vertical half-x. |
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What is the equation of the Cube-root function? Graph?
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y = ³√x or y = -³√x
The graph is a horizontal half-x. |
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What is the equation of the Circle function? Graph?
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The circle does not pass the vertical line test, so its equation is not a function.
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What is the equation of the Simi-circle function? Graph?
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Top semi-circle: y = √(r² - x²)
Bottom semi-circle: y = -√(r² - x²) Graph is a semi-circle. |
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What is the equation of the absolute value function? Graph?
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y = |x| or y = -|x|
The graph is either a V or inverted V with the vertex at the origin. |
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What is the equation for the piecewise function?
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The domain and range for the square function is:
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D = (-∞,∞)
R = [0,∞) |
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The domain and range for the square root function is:
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D = [0,∞)
R = [0,∞) |
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The domain and range of a parabola opening to the right is:
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D = [0,∞)
R = (-∞,∞) |
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The domain and range of the cube function is:
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D = (-∞,∞)
R = (-∞,∞) |
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The domain and range of the cube-root function is:
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D = (-∞,∞)
R = (-∞,∞) |
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The domain and range of the + semi-circle function is:
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it depends on the equation.
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The domain and range of the -semi-circle function is:
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it depends on the equation.
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The simplest example of a piecewise function is:
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the absolute value function.
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The domain and range of the absolute value function is:
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D = (-∞,∞)
R = [0,∞) |
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The greatest interval function is written:
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f(x) = [|x|] or f(x) = int(x)
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The greatest interval symbol is defined as:
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the largest integer that is less than or equal to x.
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[| 5.01|] =
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5, because the greatest integer that is less than or equal to 5.01 is 5.
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What is the transformation of the function: y = f(x)?
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If a, h, and k are real numbers where a ≠ 0, then y = af(x - h) + k is a transformation of the function y = f(x).
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All of the transformations of a function form a:
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family of functions
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What is the equation of the square-root family of functions?
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y = a√(x - h) + k
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What is the equation of the absolute value family of functions?
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y = a|x - h| + k
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What is the equation of the square or quadratic family?
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y = a(x - h)² + k
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Are all the graphs of a particular family of functions similar?
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yes, e.g... The graph of any function in the square or quadratic family is a parabola.
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In any family, if h is positive, then the graph moves to the:
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right
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In any family, if h is negative, then the graph moves to the:
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left
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In any family, if a is positive, then:
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the graph will open upwards.
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In any family, if a is negative, then:
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the graph will open downwards.
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In any family, if k is positive, then the graph moves:
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up
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In any family, if k is negative, then the graph moves:
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down
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The interval for the vertex of a parabola is:
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V(h,k)
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A graph is stretched when:
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a > 1.
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A graph is shrunk when:
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0 < a < 1
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If k is positive then the graph will move:
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up
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If k is negative, then the graph will move:
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down
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When graphing y = af(x - h) + k, in which order are the transformations applied?
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h - a - k. It's the same as the order of operations.
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If a transformation does not change the shape of a graph, then it is a _______ transformation.
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rigid
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If a transformation changes the shape of a graph, then it is a _______ transformation.
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non-rigid
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Which transformations are rigid and which are non-rigid?
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Translating horizontally and vertically and reflection are rigid transformations.
Stretching and shrinking are non-rigid transformations. |
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How do you translate the graph of y = f(x) left or right?
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1) Change the sign of h in y = af(x - h) + k.
2) Add the value of h to each x-coordinate of the graph y = f(x). |
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How do you reflect the graph of y = f(x)?
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Change the sign of all the y-coordinates of y = f(x) to match that of a in af(x - h) + k.
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How do you stretch or shrink the graph of y = f(x)?
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Multiply each of the y-coordinates in the graph y = f(x) by the value of a in y = af(x - h) + k.
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How do you translate the graph of y = f(x) vertically?
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Add the value of k in y = af(x - h) + k to the y-coordinates of the graph of y = f(x).
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The function f(x) = x is called the:
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identity function because the coordinates in each ordered pair are identical.
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A linear function is a transformation of the ________ function.
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identity function.
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The transformation of the identity function, f(x) = x, into the linear function is in the form:
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f(x) = mx + b where m ≠ 0.
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What if m = 0 in the linear function?
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then the function has the form f(x) = b and it is called a constant function.
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Describe an even function:
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In an even function, the graph is equidistant on each side of the y-axis; Each x-coordinate and its additive inverse will produce the same y-coordinate.
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A function is an even function IFF:
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f(-x) = f(x) for every value of x in the domain of the function f.
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A function is an odd function IFF:
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f(-x) = -f(x) for every value of x in the domain of the function.
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An odd function is symmetric about the:
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origin
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How can you tell if a function is an even or odd function?
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Replace x by -x in the formula f(x) and simplify. If f(-x) = f(x) then the graph is an even function; if f(-x) = -f(x) then the graph is an odd function.
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In general, a function with only even exponents:
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is an even function and symmetric about the y-axis.
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In general, a function with only odd exponents:
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is an odd function and symmetric about the origin.
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In general, a function with both even and odd exponents:
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has neither symmetry about the y-axis or the origin.
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Can functions have symmetry about the x-axis?
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no, if they did they would fail the vertical line test and thus not be functions.
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In the sum operation for functions, (f + g)(x) =
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f(x) + g(x)
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In the difference operation for functions, (f - g)(x) =
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f(x) - g(x)
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In the product operation for functions, (f • g)(x) =
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f(x) • g(x)
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In the quotient operation for functions, (f ÷ g)(x) =
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f(x) ÷ g(x); provided that g(x) ≠ 0.
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The domain of f + g, f - g, f • g, or f ÷ g, is:
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the intersection of the domain of f with the domain of g.
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If f and g are two functions, the composition of f and g is defined by the equation:
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(f ᐤ g)(x) = f(g(x)) provided g(x) is in the domain of f.
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If f and g are two functions, the composition of g and f is defined by the equation:
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(g ᐤ f)(x) = g(f(x)) provided f(x) is in the domain of g.
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(f ᐤ g)(x) is read:
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"f after g of x."
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In (f ᐤ g)(x) which is applied to x first:
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g is applied to x first, then f.
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What are the steps to finding the domain of the composition function: (f ᐤ g)(x) ?
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1) Find the domain of f.
2) Recognize that g(x) must be located in the domain of f. 3) Find the domain of g(x). 4) The domain or (f ᐤ g)(x) has to be smaller than the domain of g(x). |
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What are the steps to finding the inverse of a one-to-one function?
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1) Replace f(x) by y.
2) Interchange x and y. 3) Solve the equation for y. 4) Replace y by f⁻¹(x). 5) Check that the domain of f is the range of f⁻¹ and and that the domain of f⁻¹ is the range of f. |
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What is the formula for direct variation?
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y = kx
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What is the formula for indirect variation?
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y = k / x
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What is the formula for joint variation?
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y = kxz
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