Download MATH-1420 Review Concepts (Haugen) Rational Expressions

MATH-1420 Review Concepts (Haugen)
Unit 1: Equations, Inequalities, Functions, and Graphs
Rational Expressions
Determine the domain of a rational expression
Simplify rational expressions
-factor and then cancel common factors
Arithmetic operations with rational expressions
-add, subtract, multiply, and divide rational expressions
Complex rational expressions
Equations
Polynomial, radical, and absolute value equations
Equations with rational exponents
Equations that are quadratic in form
Linear Inequalities and Absolute Value Inequalities
Ways we can express the solution set of an inequality:
1. Simple or compound inequalities
2. Set-builder notation
3. Geometrically (using a number line)
4. Interval notation
Basics of Functions
Relation (a correspondence between two sets)
Domain and range of a relation
Function (a special type of relation)
Vertical Line Test
More on Functions
Identify the open intervals on which a function is increasing, decreasing, or constant
Determine the relative extrema of a function
Symmetry tests
Piecewise-defined functions
Linear Functions and Slope
Three ways to express the equation of a line:
1. Slope-Intercept Form y  mx  b
2. Point-Slope Form y  y1  m  x  x1 
3. General Form Ax  By  C
Parallel Lines (equal slopes)
Perpendicular Lines (opposite reciprocal slopes)
Average Rate of Change of a Function from x1 to x2 :
f  x2   f  x1 
x2  x1
Transformations of Functions
Rigid Transformations: horizontal, vertical, and reflection
Non-rigid Transformations: horizontal and vertical stretching/shrinking
Combinations of Functions; Composite Functions
Sum, Difference, Product, and Quotient Functions
Function Composition
Inverse Functions
One-to-One Functions and the Horizontal Line Test
Finding the inverse of a function (Switch-and-Solve Approach)
Unit 2: Polynomial and Rational Functions
Quadratic Functions
Standard Form: f  x   a  x  h   k
2
General Form: f  x   ax2  bx  c
Identify the vertex, axis of symmetry, x-intercept(s), and the y-intercept of a parabola
Polynomial Functions
General Form: p  x   an xn  an1 xn1  an2 xn2  ...  a2 x2  a1x  a0
End behavior of a polynomial function
Locating the zeros of a polynomial function
Determine the behavior of a polynomial function at its zeros
Intermediate Value Theorem
Dividing Polynomials
Polynomial long division
Synthetic division
Zeros of Polynomial Functions
Rational Zero Theorem
Conjugate Pairs Theorem
Descartes’ Rule of Signs
Graphs of Rational Functions
Asymptotic behavior of rational functions
Vertical, Horizontal, and Slant (or Oblique) Asymptotes
Polynomial and Rational Inequalities
Boundary points, test intervals, test values, endpoint analysis, and conclusion
Unit 3: Exponential and Logarithmic Functions
Exponential Functions
Standard Form: f  x   b x , b  0, b  1
Properties of exponential functions
Compound Interest Formulas:
nt
 r
Finite number of compounding periods: A  P  1  
 n
Infinite number of compounding periods (continuous compounding): A  Pe rt
Logarithmic Functions
Standard Form: f  x   logb  x  , b  0, b  1
Evaluating logarithms
Converting from logarithmic form to exponential form and vice versa
Properties of logarithms
Product Rule (the log of the product = the sum of the logs)
Quotient Rule (the log of the quotient = the difference of the logs)
Power Rule (special case of the Product Rule)
log a  M 
Change of Base Formula: log b  M  
log a  b 
Exponential and Logarithmic Equations
“Type 1” exponential equations
Express each side using the same base and then use 1-1 property of exponential functions
“Type 2” exponential equations
Take the logarithm of each side and then apply the Power Rule
“Type 1” logarithmic equations
Convert to exponential form and then solve
“Type 2” logarithmic equations
Take advantage of the Product, Quotient, and/or Power Rule
Watch for extraneous solutions
Exponential Growth and Decay; Modeling Data
Equation: A  A0e kt
We have exponential growth when k  0 ; decay when k  0
A0 is the original amount or size of the growing/decaying entity
Logistic Growth
Growth under restricted conditions
c
Equation: A 
1  ae  bt
Unit 4: Conic Sections
Distance and Midpoint Formulas
Distance between the points  x1 , y1  and  x2 , y2  :
d
 x2  x1    y2  y1 
2
2
Midpoint of the line segment joining  x1 , y1  and  x2 , y2  :
 x  x y  y2 
Midpoint   1 2 , 1

2 
 2
Circles
Standard form of the equation of a circle:
 x  h   y  k 
2
2
 r2
General form of the equation of a circle: x2  y 2  Dx  Ey  F  0
Convert from general form to standard form by completing the square
Ellipses
Standard form of an ellipse centered at the origin (assume a  b ):
x2 y 2

1
horizontal major axis
a 2 b2
x2 y 2

1
vertical major axis
b2 a 2
Standard form of an ellipse centered at the point  h, k  :
 x  h
2
a2
 x  h
 y k

2
 y k

2
1
b2
2
horizontal major axis
1
vertical major axis
b2
a2
Identify center, vertices, foci, and the endpoints of the minor axis
Use c 2  a 2  b 2 to help locate the foci
General form of the equation of an ellipse: Ax2  By 2  Dx  Ey  F  0,  A and B  0
Convert to standard form by completing the square
Hyperbolas
Standard form of a hyperbola centered at the origin:
x2 y 2

1
horizontal transverse axis
a 2 b2
y 2 x2
 1
vertical transverse axis
a 2 b2
Standard form of a hyperbola centered at the point  h, k  :
 x  h
2
a2
y k

2
 x  h

2
1
b2
 y k
2
horizontal transverse axis
1
vertical transverse axis
a2
b2
Identify center, vertices, foci, and the endpoints of the conjugate axis
Use c 2  a 2  b 2 to help locate the foci
Fundamental Rectangle helps us sketch each branch
Equations of the asymptotes
General form of the equation of a hyperbola: Ax2  By 2  Dx  Ey  F  0,  A and B  0
Convert from general form to standard form by completing the square
Parabolas
Standard form of a parabola whose vertex is located at the origin:
x 2  4 py
opens up or down
y 2  4 px
opens left or right
Focal length, p, is the directed distance from the vertex to the focus of the parabola
Standard form of a parabola with vertex  h, k  :
 x  h  4 p  y  k 
2
 y  k   4 p  x  h
2
opens up or down
opens left or right
Identify the vertex, focus, and directrix of a parabola
Latus rectum helps us sketch parabolas
Length of a parabola’s latus rectum = 4 p
General form of the equation of a parabola: Ax2  By 2  Dx  Ey  F  0,
Convert to standard form by completing the square
 A or B  0
Unit 5: Systems of Equations/Inequalities and Matrices
Systems of Linear Equations in Two Variables
A solution to a system must satisfy all equations simultaneously
Solve by graphing, substitution, or elimination
Recognize when a system has an infinite number of solutions
Recognize when a system has no solutions
Systems of Linear Equations in Three Variables
To solve a system of three equations with three unknowns:
1. Pick two equations and eliminate an unknown
2. Pick another two equations and eliminate the same unknown from step 1
3. Solve the 2x2 system formed using the equations from steps 1 and 2
Recognize when a system has an infinite number of solutions
Recognize when a system has no solutions
Systems of Nonlinear Equations in Two Variables
We can use the elimination method is some cases; otherwise, substitution must be used
Watch for extraneous solutions
It helps to sketch each equation in the system
Systems of Inequalities
For each inequality in the system:
1. Sketch the boundary line (dashed or solid depending on the inequality)
2. Pick a test point not on the boundary line
3. Shade on the appropriate side of the boundary line
The solution to the system is given by the overlapping shaded regions (assuming such an overlap exists)
Matrix Solutions to Linear Systems
Augmented Matrix
Elementary Row Operations
Row-Echelon Form
We convert the augmented matrix to row-echelon form using Gaussian elimination
Reduced Row-Echelon Form
Gauss-Jordan elimination
Recognize when a system has an infinite number of solutions
Recognize when a system has no solutions
Matrix Operations
Matrix addition and subtraction
Scalar multiplication
Matrix multiplication