LSU
Mathematics

# Math 1023 Algebra and Trigonometry Topics

## Math 1023 Pages

Linear Equations

• Recognizing linear equations
• Solving linear equations with integer coefficients
• Solving linear equations involving fractions
• Solving linear equations involving decimals
• Solving equations that lead to linear equations

Quadratic Equations

• Solving quadratic equations by factoring and the zero product property
• Solving quadratic equations using the square root property
• Solving quadratic equations using the quadratic formula
• Using the discriminant to determine the type of solutions of a quadratic equation

Other Types of Equations

• Solving higher-order polynomial equations
• Solving equations that are quadratic in form
• Solving equations involving single radicals

Linear Inequalities

• Solving linear inequalities
• Solving three-part inequalities

Absolute Value Equations and Inequalities

• Solving an absolute value equation
• Solving an absolute value “less than” inequality
• Solving an absolute value “greater than” inequality

The Rectangular Coordinate System

• Plotting ordered pairs
• Finding the midpoint of a line segment using the midpoint formula
• Finding the distance between two points using the distance formula

Circles

• Writing the standard form of an equation of a circle
• Sketching the graph of a circle
• Converting the general form of a circle into standard form

Lines

• Determining the slope of a line
• Sketching a line given a point and the slope
• Finding the equation of a line using the point-slope form
• Finding the equation of a line using the slope-intercept form
• Writing the equation of a line in standard form
• Finding the slope and the y-intercept of a line in standard form
• Sketching lines by plotting intercepts
• Finding the equation of a horizontal line and a vertical line

Parallel and Perpendicular Lines

• Understanding the definition of parallel lines
• Understanding the definition of perpendicular lines
• Determining whether two lines are parallel, perpendicular, or neither
• Finding the equations of parallel and perpendicular lines

Relations and Functions

• Understanding the definitions of relations and functions
• Determining whether equations represent functions
• Using function notation; evaluating functions
• Using the vertical line test
• Determining the domain of a function given the equation

Properties of a Function’s Graph

• Determining the intercepts of a function
• Determining the domain and range of a function from its graph
• Determining whether a function is increasing, decreasing, or constant
• Determining relative maximum and relative minimum values of a function
• Determining whether a function is even, odd, or neither
• Determining information about a function from a graph

Graphs of Basic Functions; Piecewise Functions

• Sketching the graphs of the basic functions
• Analyzing piecewise-defined functions

Transformations of Functions

• Using vertical shifts to graph functions
• Using horizontal shifts to graph functions
• Using reflections to graph functions
• Using vertical stretches and compressions to graph functions
• Using combinations of transformations to graph functions

Composite Functions

• Forming and evaluating composite functions
• Determining the domain of composite functions

One-to-One Functions; Inverse Functions

• Understanding the definition of a one-to-one function
• Determining whether a function is one-to-one using the horizontal line test
• Understanding and verifying inverse functions
• Sketching the graphs of inverse functions
• Finding the inverse of a one-to-one function

Quadratic Functions

• Understanding the definition of a quadratic function and its graph
• Graphing quadratic functions written in standard form
• Graphing quadratic functions using the vertex formula
• Determining the equation of a quadratic function given its graph

Applications of Quadratic Functions

• Maximizing projectile motion functions
• Maximizing functions in economics

The Graphs of Polynomial Functions

• Understanding the definition of a polynomial function
• Sketching the graphs of power functions
• Determining the end behavior of polynomial functions
• Determining the intercepts of a polynomial function
• Determining the real zeros of polynomial functions and their multiplicities
• Sketching the graph of a polynomial function
• Determining a possible equation of a polynomial function given its graph

Rational Functions and Their Graphs

• Finding the domain and intercepts of rational functions
• Identifying vertical asymptotes
• Identifying horizontal asymptotes
• Using transformations to sketch the graphs of rational functions
• Sketching rational functions
• Sketching rational functions having removable discontinuities

Exponential Functions

• Understanding the characteristics of exponential functions
• Sketching the graphs of exponential functions using transformations
• Solving exponential equations by relating the bases
• Solving applications of exponential functions

Logarithmic Functions

• Understanding the definition of a logarithmic function
• Evaluating logarithmic expressions
• Understanding the properties of logarithms
• Using the common and natural logarithms
• Understanding the characteristics of logarithmic functions
• Sketching the graphs of logarithmic functions using transformations
• Finding the domain of logarithmic functions

Properties of Logarithms

• Using the product rule, quotient rule, and power rule for logarithms
• Expanding and condensing logarithmic expressions
• Solving logarithmic equations using the logarithm property of equality
• Using the change of base formula

Exponential and Logarithmic Equations

• Solving exponential equations
• Solving logarithmic equations

Applications of Exponential and Logarithmic Functions

• Solving compound interest applications
• Solving exponential growth and decay applications

An Introduction to Angles: Degree and Radian Measure

• Understanding degree measure and radian measure
• Converting between degree measure and radian measure
• Finding coterminal angles using degree measure and radian measure

Applications of Radian Measure

• Determining the area of a sector of a circle
• Determining the arc length of a sector of a circle

Triangles (Review)

• Classifying triangles
• Using the Pythagorean Theorem
• Understanding similar triangles
• Understanding the special right triangles

Right Triangle Trigonometry

• Understanding the right triangle definitions of the trigonometric functions
• Using the special right triangles
• Understanding the fundamental trigonometric identities
• Understanding cofunctions
• Evaluating trigonometric functions using a calculator

Trigonometric Functions of General Angles

• Understanding the four families of special angles
• Understanding the definitions of the trigonometric functions of general angles
• Finding the values of the trigonometric functions of quadrantal angles
• Understanding the signs of the trigonometric functions
• Determining reference angles
• Evaluating trigonometric functions of angles belonging to the $\frac{\pi}{3}$, $\frac{\pi}{4}$, and $\frac{\pi}{6}$ families

The Unit Circle

• Understanding the definition of the unit circle
• Understanding the unit circle definitions of the trigonometric functions

The Graphs of the Trigonometric Functions

• Understanding the graphs of the sine, cosine, tangent, cotangent, secant, and cosecant functions and their properties
• Sketching graphs of the form $y=A\sin(Bx-C)+D$ or $y=A\cos(Bx-C)+D$
• Sketching graphs of the form $y=A\tan(Bx-C)+D$ or $y=A\cot(Bx-C)+D$
• Sketching graphs of the form $y=A\sec(Bx-C)+D$ or $y=A\csc(Bx-C)+D$
• Determine the equation of a function of the form y=Asin(Bx-C) or y=Acos(Bx-C) given its graph

Inverse Trigonometric Functions

• Understanding and finding the exact and approximate values of the inverse sine function, the inverse cosine function, and the inverse tangent function
• Evaluating composite functions involving inverse trigonometric functions of the forms $f∘f^{-1}$, $f^{-1}∘f$, $f∘g^{-1}$, and $f^{-1}∘g$

Trigonometric Identities

• Substituting known identities to verify an identity
• Changing to sines and cosines to verify an identity
• Factoring to verify an identity
• Separating a single quotient into multiple quotients to verify an identity
• Combining fractional expressions to verify an identity
• Multiplying by conjugates to verify an identity

The Sum and Difference Formulas

• Understanding and using the sum and difference formulas for the cosine, sine, and tangent functions
• Using the sum and difference formulas to verify identities
• Using the sum and difference formulas to evaluate expressions involving inverse trigonometric functions

The Double-Angle and Half-Angle Formulas

• Understanding and using the double-angle formulas and the half-angle formulas
• Using the double-angle and half-angle formulas to verify identities
• Using the double-angle and half-angle formulas to evaluate expressions involving inverse trigonometric functions

Trigonometric Equations

• Solving trigonometric equations that are linear or quadratic in form
• Solving trigonometric equations using identities
• Solving other types of trigonometric equations
• Solving trigonometric equations using a calculator

Right Triangle Applications

• Solving right triangles
• Solving applications using right triangles

The Law of Sines

• Determining if the Law of Sines can be used to solve an oblique triangle
• Using the Law of Sines to solve the SAA case or the ASA case
• Using the Law of Sines to solve the SSA (ambiguous) case
• Using the Law of Sines to solve applied problems involving oblique triangles

The Law of Cosines

• Determining if the Law of Cosines can be used to solve an oblique triangle
• Using the Law of Cosines to solve the SAS case
• Using the Law of Cosines to solve the SSS case
• Using the Law of Cosines to solve applied problems involving oblique triangles

Area of Triangles

• Determining the area of oblique triangles
• Using Heron’s Formula to determine the area of an SSS triangle
• Solving applied problems involving the area of triangles

Polar Coordinates and Polar Equations

• Plotting points using polar coordinates
• Determining different representations of a point $(r, θ)$
• Converting points from polar to rectangular coordinates and from rectangular to polar coordinates
• Converting equations from rectangular to polar form and from polar to rectangular form

Graphing Polar Equations

• Sketching equations of the form $r\cosθ = a$, $r\sinθ = a$, $ar\cosθ + br\sinθ = c$, and $θ = α$
• Sketching equations of the form $r = a$, $r = a\sinθ$, and $r = a\cosθ$
• Sketching equations of the form $r = a + b\sinθ$ and $r = a + b\cosθ$
• Sketching equations of the form $r = a\sin(nθ)$ and $r = a\cos(nθ)$
• Sketching equations of the form $r^{2} = a^{2}\sin(2θ)$ and $r^{2} = a^{2}\cos(2θ)$

Vectors

• Understanding the geometric representation of a vector
• Understanding operations on vectors represented geometrically
• Understanding vectors in terms of components
• Understanding vectors in terms of i and j
• Finding a unit vector
• Determining the direction angle of a vector
• Representing a vector in terms of i and j given its magnitude and direction angle
• Using vectors to solve applied problems involving velocity

Systems of Equations

• Verifying solutions to a system of linear equations in two variables
• Solving a system of linear equations using the substitution method
• Solving a system of linear equations using the elimination method
• Solving applied problems using a system of linear equations