MATH-1420 Review Concepts (Haugen) Rational Expressions

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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

   

1

3.

General Form Ax

By

C

Parallel Lines (equal slopes)

Perpendicular Lines (opposite reciprocal slopes)

Average Rate of Change of a Function from x

1

to x

2

:

     x

2

 x

1

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:

General Form:

     2  k

   ax

2 bx c

Identify the vertex, axis of symmetry, x -intercept(s), and the y -intercept of a parabola

Polynomial Functions

General Form:

   a x n  a x n

1  a n

2 x n

2

...

a x

2  a x a

0

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:

   b x

, b

0, b

1

Properties of exponential functions

Compound Interest Formulas:

Finite number of compounding periods: A

P

1

 r nt n

Infinite number of compounding periods (continuous compounding): A

Pe rt

Logarithmic Functions

Standard Form:

 log b

 

, 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)

Change of Base Formula: log

 log a

  b log a

M

 

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

A e kt

We have exponential growth when k

0 ; decay when k

0

A

0

is the original amount or size of the growing/decaying entity

Logistic Growth

Growth under restricted conditions

Equation: A

1

 c ae

 bt

Unit 4: Conic Sections

Distance and Midpoint Formulas

Distance between the points

 x y

1

and

 x y

2

: d

  x

2

 x

1

  y

2

 y

1

2

Midpoint of the line segment joining

 x y

1

and

 x y

2

:

Midpoint x

1

2 x

2 , y

1

2 y

2

Circles

Standard form of the equation of a circle:

 x h

  y

 k

 2  r

2

General form of the equation of a circle: x

2  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 ): x a

2

2 y b

2

 

2

1

    a

2 b

2

2

1 horizontal major axis x b

2

2 y a

2

 

2

1 vertical major axis

Standard form of an ellipse centered at the point

 

: horizontal major axis

    b

2 a

2

2

1 vertical major axis

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: Ax

2 

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: x a

2

2 y b

2

 

2

1

   

2 a

2 b

2

1 horizontal transverse axis y a

2

2 x b

2

 

2

1 vertical transverse axis

Standard form of a hyperbola centered at the point

 

: horizontal transverse axis

   

2 a

2 b

2

1 vertical transverse axis

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: Ax

2

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

 

:

 x h

 2 

4

  k

 y

 k

2 

4

  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 =

General form of the equation of a parabola:

4 p

Ax

2 

By

2 

Dx Ey F 0,

A or B

0

Convert to standard form by completing the square

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

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