Math 1023 Algebra and Trigonometry Topics

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