# Shared Flashcard Set

## Details

Algebra Formulas
(Midterm) Formulas having to do with functions mostly
67
Mathematics
02/15/2010

Term
 Looking at a graph, how can you tell if it is a function?
Definition
 Vertical Line Test:   Vertical line should not intersect graph in more than one place. If it does, then it is not a function.
Term
 The output variable and the input variable:   Which one is dependent, and which is independent?
Definition
 The output variable is dependent (it depends upon the input)   The input variable is independent (may have its value freely chosen regardless of any other variable values)     The output is a function of (depends upon) the input
Term
 The DOMAIN (or INPUT) is on the _____ axis.   The RANGE (or OUTPUT) is on the _____ axis.
Definition
 The domain is x (x-axis)   The range is f(x) or y (y-axis)
Term
 Pythagorean Theorem for a Right Triangle
Definition
 L2 + H2 = D2   L = Length H = Height D = Diagonal
Term
 Area of a Circle
Definition
 (pi) r2
Term
 Area of a Triangle
Definition
 (B x H) --------- 2
Term
 Vertical Shift of Function (up/down):   Horizontal Shift of Function (left/right):
Definition
 Vertical Shift of Function (up/down): Add or subtract from the function EG: f(x) → f(x) + 5 will move up 5 units   Horizontal Shift of Function (left/right): Add or subtract the reverse from x EG: f(x) → f(x-5) will move right 5 units
Term
 Reflect Function Across x-axis:   Reflect Function Across y-axis:
Definition
 Reflect Function Across x-axis: Multiply function by -1 EG: f(x) → -f(x) will mirror across x-axis   Reflect Function Across y-axis: Multiply x by -1 EG: f(x) → f(-x) will mirror across y-axis
Term
 Vertically Stretch Graph of a Function:   Vertically Shrink Graph of a Function:
Definition
 Vertically Stretch Graph of a Function: Multiply function by a number greater than 1 EG: f(x) → 3f(x) will vertically stretch the graph   Vertically Shrink Graph of a Function: Multiply function by a number between 0 and 1 EG: f(x) → 0.5f(x) will vertically shrink the graph
Term
 Horizontally Stretch Graph of a Function:   Horizontally Shrink Graph of a Function:
Definition
 Horizontally Stretch Graph of a Function: Multiply x by a number between 0 and 1 EG: f(x) → f(0.5x) will horizontally stretch the graph   Horizontally Shrink Graph of a Function: Multiply x by a number greater than 1 EG: f(x) → f(3x) will horizontally shrink the graph
Term
 Odd and Even Functions
Definition
 f(x) = f(-x) is EVEN (symmetry about the y-axis)   f(x) = -f(x) not possible except for 0 (symmetry about the x-axis)   -f(x) = f(-x) and f(-x) = -f(x) are ODD (symmetry about the origin)
Term
 How to find the inverse of a function:
Definition
 1. Replace f(x) with y   2. Solve for x in terms of y (x on one side, alone)   3. Interchange x and y, then replace y with f-1(x)
Term
 Is this a function?   X: 3, 2, 4, 6, 8, 12 Y: 3, 3, 7, 12, 4, 8
Definition
 Yes - passes Vertical Line Test   All Domain values are unique
Term
 Is this a function?   X: 3, 2, 4, 3, 8, 12 Y: 3, 3, 7, 12, 4, 8
Definition
 No - does not pass Vertical Line Test   Domain contains duplicates (3 corresponds to two values in the range- 3 and 12)
Term
 Slope of a Linear Function in terms of Rise and Run
Definition
 Rise Slope = ----------           Run
Term
 Standard Form of a Linear Function
Definition
 y or f(x) = mx + b   m is the slope   b is the y-intercept
Term
 How to calculate slope from coordinates of 2 points on the line:
Definition
 For (x1, y1) (x2, y2)        y2 - y1 M = -----------      x2 - x1
Term
 Point-Slope Form
Definition
 y or f(x) = m(x-x1) + y1   (x-x1) ends up being x   y1 ends up being b or y-intercept
Term
 How to find the root of a linear function:
Definition
 Calculate y = mx + b as 0 = mx + b
Term
 Parallel lines have slopes that are ______   Perpendicular lines have slopes that are ______
Definition
 Parallel lines have slopes that are EQUAL EG: m1 = m2   Perpendicular lines have slopes that are NEGATIVELY RECIPROCAL EG: m1 = -1/m2   or   m2 = -1/m1
Term
 How to find the point of intersection of 2 lines:
Definition
 For two lines     y1=m1x1+b1    and    y2=m2x2+b2                                  b2-b1 Point of intersection (x0) is ------------                                m1-m2     (then can use this as x to find y)
Term
 In regression analysis,   r is ________   and r2 is ________
Definition
 r is the CORRELATION COEFFICIENT (a number between -1 and 1 that measures how well the best fitting line fits the data points)   r2 is the COEFFICIENT OF DETERMINATION (a number that determines if the best fitting line can be used as a data model. Closer to 1, the better the fit)
Term
 Standard form of a Quadratic Function
Definition
 y or f(x) = ax2+bx+c   (a≠0, if a=0 then it is a horizontal line)
Term
 Standard Form vs. Vertex Form of a Quadratic Function
Definition
 Standard Form y or f(x) = ax2+bx+c   Vertex Form y or f(x) = a(x-h)2+k   -h,k are the x,y of the vertex
Term
 Move the vertex of   f(x) = 3x2+1 to (-3,2)
Definition
 Replace old x with new x 3x2 → 3(x+3)2 remember, x moves the opposite way   Replace old y with new y +1 → +2   3x2+1 → 3(x+3)2 +2
Term
 Finding the Vertex of a Quadratic Function
Definition
 -b x = ------      2a     Plug this into the equation to find y
Term
 Finding the roots of a Quadratic Function:
Definition
 The root(s) are at 0 = ax2+bx+c   Use the Quadratic Formula:       -b ± √b2 - 4ac x= --------------------     2a
Term
 What is the Quadratic Formula?   What is it used for?
Definition
 Quadratic Formula       -b ± √b2 - 4ac x= --------------------     2a   Quadratic Formula is used to find the roots of a quadratic function
Term
 What is the Discriminant and what can it tell you?
Definition
 The Discriminant is the b2 - 4ac part of the Quadratic Function   If the Discriminant is positive, there are two roots   If the Discriminant is zero, there is one root, the graph is sitting on the x-axis   If the Discriminant is negative, the graph does not intersect the x-axis (there is no root)
Term
 What kind of function is this:   f(x) = mx + b
Definition
 Linear Function
Term
 Linear Regression Analysis   What is the correlation coefficient and how is it represented?
Definition
 correlation coefficient = r   Measures how well the best fitting line fits the data points. Ranges from -1 to 1.
Term
 Linear Regression Analysis   What is the coefficient of determination and how is it represented?
Definition
 Coefficient of Determination = r2 (the square of the correlation coefficient). Determines if the best fitting line can be used as a model (is it good enough?)   The closer r2 is to 1, the better the fit.
Term
 What kind of function is this:   f(x) = ax2 + bx + c
Definition
 Quadratic Function   (a ≠ 0)   The simplest form of a quadratic function is f(x) = x2   a.k.a. a power function
Term
 What kind of function is this:   ax4 + ax3 + ax2 + ax + a
Definition
 Polynomial Function   (of degree 4 - quartic polynomial)
Term
 Standard form of a Polynomial Function
Definition
 ax4 + ax3 + ax2 + ax + a   (the exponent cannot be negative, the exponent cannot be a fraction, x cannot be in the denominator)
Term
 If the first (largest) term in a polynomial function is   ax4 the function is ____________   ax3 the function is ____________   ax2 the function is ____________   ax the function is  ____________   ax0 ________________
Definition
 If the first (largest) term in a polynomial function is   ax4 the function is quartic (parabola) ax3 the function is cubic (snakelike) ax2 the function is quadratic (parabola) ax the function is linear (line) ax0 is a horizontal line at y=a
Term
 Polynomial Function   bx4 + ax3 + ax2 + ax + g   What is b? What is 4? What is g? What is bx4?
Definition
 b is the leading coefficient 4 is the degree/order g is the constant term bx4 is the leading term
Term
 f(x) = axn
Definition
 f(x) = axn   is a monomial function is a power function   (n > 0 b ≠ 0)
Term
 f(x) = axn   if n=0, graph is _________________   if n=1, graph is _________________   if n=2, graph is _________________   if n=3, graph is _________________
Definition
 f(x) = axn   if n=0, graph is a horizontal line at y=a   if n=1, graph is linear with slope of a (odd function)   if n=2, graph is parabola, branches facing up when a is a is positive, down when a is negative (even function)   if n=3, graph is snakelike, increasing when a is positive, decreasing when a is negative (odd function)
Term
 Even-exponent Power Functions   xn → n could equal _____   the shape is _______   graph gets ______ the _______ the exponent   When x>1 or x<-1, ______ are ________   When -1>x>1, _______ are ________
Definition
 xn → n could equal 2, 4, etc.   the shape is a parabola   graph gets flatter (on the bottom) the higher the exponent   When x>1 or x<-1, branches are steeper   When -1>x>1, branches are flatter
Term
 Odd-exponent Power Functions   xn → n could equal _____   the shape is _______   graph gets ______ the _______ the exponent   When x>1 or x<-1, ______ are ________   When -1>x>1, _______ are ________
Definition
 xn → n could equal 1, 3, 5, etc.   the shape is snakelike   graph gets flatter (on the bottom) the higher the exponent   When x>1 or x<-1, traces are steeper   When -1>x>1, traces are flatter
Term
 Intermediate Value Theorem   (polynomial functions)
Definition
 If the result of f(a) and f(b) are opposite signs (+/-), then there must be at least one root between them   (as long as a≠b)
Term
 Factor Theorem   (polynomial functions)
Definition
 f(c) will equal zero ONLY IF (x-c) is a factor of the polynomial.   In other words, the factors (x-c) are the only places where the function will equal zero.
Term
 i2 =   √-16 =
Definition
 i2 = -1   √-16 = √16 i = 4i
Term
 (x-c)3 has a _________ of _____   if x=4, the factor of the polynomial is ______   if x = -3, the factor of the polynomial is ______
Definition
 (x-c)3 has a multiplicity of 3   if x=4, the factor of the polynomial is (x-4)   if x = -3, the factor of the polynomial is (x+3)
Term
 (x-c)3 will _____ the x-axis at the x=c   (x-c)2 will _____ the x-axis at the x=c
Definition
 (x-c)3 will cross the x-axis at the x=c   (x-c)2 will touch the x-axis at the x=c
Term
 How do you represent a polynomial factor that does not cross or touch the x-axis anywhere?
Definition
 The constant factor k   f(x) = k(x-c1)(x-c2)(x-c3)   Adding or subtracting from the constant factor k shifts the graph up or down the y-axis
Term
 A polynomial of degree/order "n" can have a maximum of ___ roots   A polynomial of degree/order "n" can have a maximum of ___ turning points
Definition
 A polynomial of degree/order "n" can have a maximum of n roots   A polynomial of degree/order "n" can have a maximum of n-1 turning points
Term
 Finding the rational (not irrational) zeros of a polynomial function:   Rational Zeros Theorem
Definition
 p (all rational factors of constant term)      ___                                                            r (all rational factors of leading coefficient)   any of these that lead to f(x)=0 are the rational zeros
Term
 A quadratic function can have ____ turning points   A cubic function can have ____ turning points   A quartic function can have ____ turning points
Definition
 A quadratic function can have 1 turning point   A cubic function can have 2 turning points   A quartic function can have 3 turning points
Term
 Polynomial Functions   When the absolute value of x is large, end/long-run behavior of the graph will tend to ______
Definition
 When the absolute value of x is large, end/long-run behavior of the graph will tend to follow the leading term
Term
 For every polynomial function where the degree is >0, there are complex numbers such that f(x)=a(x-c1) (x-c2)... etc...   (as long as a≠0)   This is known as _____________
Definition
 The Linear Factorization Theorem     For every polynomial function where the degree is >0, there are complex numbers such that f(x)=a(x-c1) (x-c2)... etc...   (as long as a≠0)
Term
 Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.   This is called _______________
Definition
 The Fundamental Theorem of Algebra     Every polynomial of a degree of ≥1 with complex coefficients has at least one zero in the complex number system.
Term
 What kind of function is this:           p(x) f(x)= -----------         q(x)
Definition
 Rational Function
Term
 What kind of function is this:           ax3+bx2+cx+d f(x)= -------------------------         ax4+bx3+cx2+dx+e
Definition
 Rational Function
Term
 What is the domain of a rational function?
Definition
 The domain of a rational function is the set of all real numbers that are NOT roots of the denominator (the denominator≠0)
Term
 (x-2)(x+6)2 ---------------- (x-2)(x-6)   1. Root(s)/Zero(s): 2. Vertical Asymptote(s): 3. Hole(s): 4. Degree of numerator/denominator:
Definition
 (x-2)(x+6)2 ---------------- (x-2)(x-6)   1. Root(s)/Zero(s): -6 2. Vertical Asymptote(s): 6 3. Hole(s): 2 4. Degree of numerator/denominator: 3/2
Term
 x3+10x2+12x-72 ----------------------- x2-8x+12   1. Y-intercept: 2. Horizontal Asymptote(s): 3. End behavior of graph: 4. Degree of numerator/denominator:
Definition
 x3+10x2+12x-72 ----------------------- x2-8x+12   1. Y-intercept: x=0 is not a root of the denominator, so evaluate function at x=0. y=-6 2. Horizontal Asymptote(s): oblique asymptote, divide the equation to find it. x+18 3. End behavior of graph: x3/x2 which would be a line increasing as x increasing that crosses the graph at x=2 and x=6 4. Degree of numerator/denominator: 3/2
Term
 Negative or positive? What degree?   [image]
Definition
 Negative   Odd
Term
 Negative or positive? What degree?   [image]
Definition
 Positive   Odd
Term
 Negative or positive? What degree?   [image]
Definition
 Negative   Even
Term
 Negative or positive? What degree?  [image]
Definition
 Positive   Even
Term
 Asymptote of   an ------- bn
Definition
 Horizontal asymptote at   y=a/b
Term
 Asymptote of   an ------ bN
Definition
 Horizontal asymptote at   y=0
Term
 Asymptote of   aN ------ bn
Definition
 Oblique asymptote at   (divide the equation to find it)
Term
 Asymptote of   aNN --------- bn
Definition
 No line asymptote
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