# 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