MA001: College Algebra (2022.A.01)

This passage assumes you have already been exposed to solving a variety of linear equations. This refresher is intended to support you as you explore and manipulate linear functions.

This section provides you with applications of the linear equation and its representation on the Cartesian plane. Examples are given in the context of real-world models and scenarios.

This section will prepare you for studying rational functions and their graphs. Methods for solving rational equations include using the LCD and factoring. Examples also include determining when there are excluded values in the solution.

This summary of algebraic operations on complex numbers will prepare you for solving quadratic equations with no solutions and the related implications for graphing quadratic and polynomial functions.

This is a refresher on solving quadratic equations using factoring methods. This section assumes you have been exposed to methods for factoring previously.

This is a refresher on solving quadratic equations using the square root property. This section assumes you have been exposed to simplifying roots algebraically.

This is a refresher on using the quadratic formula to solve quadratic equations. This section will introduce the discriminant and explain how to use it to classify the number and type of solutions to a quadratic equation. This analysis is an essential step in learning how to analyze the behavior of functions using algebraic and graphical methods.

This section explores solving different types of equations and analyzing solutions.

In this section, you will explore methods for solving absolute value equations and how to analyze solutions to determine their feasibility. This section will help you become familiar with the algebra of absolute value equations in preparation for functions.

This is a refresher on solving various radical equations and determining whether there are extraneous solutions. Radical is another term for root, so you will be solving equations that contain square roots.

This section is an important foundation for learning how to express intervals and sets fluently using words, set-builder notation, and interval notation. You will use the concepts you learn here to describe the solutions to inequalities. Similarly, we can use sets to describe the behavior of all types of functions. You will be required to use the notation and concepts presented here in many future units in the course, so it is vital to make sure that you achieve mastery of the techniques presented here to be successful in the coming sections on functions.

This refresher on solving linear inequalities allows you to practice describing solutions using interval notation, set notation, and graphs. You will also have a chance to practice solving compound linear inequalities.

This refresher on solving absolute value inequalities lets you practice writing solutions using interval notation. You will also practice graphical analysis of absolute value inequalities.

This section will introduce the cartesian coordinate plane and how to plot points and lines on it. You will also evaluate a two-variable linear equation and learn how a point on the cartesian plane can be a solution to a linear equation in two variables.

This section introduces the terminology and notation used to define and represent a function using words, function notation, and tables. We will use the concepts and notation introduced in this section throughout the course, so make sure you master them before moving on.

In this section, you will analyze graphs to determine whether they represent a function and be introduced to the graphs of the basic functions. Pay close attention to the basic functions because they will be referred to throughout most of the course.

In this section, we will find the domain and range of functions given their graphs.

We can apply the domain and range of functions in piecewise-defined functions. These functions are defined in pieces, and understanding how to define their domains helps us understand their behavior and how to define and describe them.

We will continue exploring functions using equations, tables, words, and graphs by finding average rates of change of functions.

Composite functions are combinations of functions. This section will teach you how to manipulate composite functions algebraically, define their domain and range, and graph them.

In this section, you will learn how to define the domain of a composite function.

The next transformation we will learn is reflections. You will reflect graphs of functions across the axes and determine how the transformations change the equations of the functions.

Graphs can give us important clues about the behavior and characteristics of functions. In this section, you will explore the idea of a one-to-one function. This is an important concept for defining the inverse of a function. If you move on to calculus, this is an important concept that will return when you learn about rates of change and derivatives.

The last set of transformations you will explore on the graphs of functions are stretches and compressions. You will find out how the equation of a function changes when you stretch the graph in the x-direction or the y-direction.

Inverse functions bring together the concepts we have learned about composite functions and domain and range, so make sure you are confident with those sections before you dig in. A function must be one-to-one to have an inverse. If you are uncertain how to determine whether a function is one-to-one, it may be a good idea to revisit that concept.

In this introductory section, you will practice representing a linear function using words, function notation, tables, and graphs. We will also introduce the language and techniques used to determine whether the graph of a linear function is increasing, decreasing, or constant.

Now, you will build on your knowledge of writing equations of linear functions, drawing their graphs, and performing transformations by exploring parallel and perpendicular lines. You will learn how to identify whether lines are parallel or perpendicular given their equations and graphs. You will also learn how to define the equation of a line parallel or perpendicular to another line given an equation and a point.

This section introduces a basic problem-solving method that will help you navigate the task of creating a linear model given verbal descriptions.

Now, we will look at a system of two linear equations. There are three different types of solutions for a system of linear equations. We will explore this topic in-depth later in the course.

We will continue our study of functions by exploring the characteristics of quadratic functions. You may have solved quadratic equations in the past, and now we will bring together the quadratic equation and the quadratic function. We will explore the graph of a quadratic function and use some of the same techniques for solving quadratic equations to find special points on the function. You will learn how to identify the vertex, axis of symmetry, and intercepts of the graph of a quadratic function and how to calculate their value given the equation of a quadratic function.

The domain and range of a function are integral to its definition. In this section, you will learn how to use algebraic techniques to define a function's domain and range given its equation.

You will use the ideas in this section to define key characteristics of polynomial and rational graphs. These ideas continue into first-year calculus and help us analyze behaviors of functions and trends in general.

In this section, you will learn how to identify a power function and use interval notation to express its long-run behavior. If you need a refresher on how to use interval notation, now is a good time to review.

Now, you will learn how to identify a polynomial function and what makes them different from a power function. You will also be able to define the key characteristics of a polynomial function, such as the degree, leading coefficient, end behavior, intercepts, and turning points.

This section will dig deeper into the relationship between the graph of a polynomial function and its equation. You will see how to use the factors of a polynomial function to determine where the x-intercepts are, and you will also learn about the multiplicity of a zero (x-intercept) and how to find it.

In this section, we will bring all we know about polynomial functions and use it to sketch a graph given an equation. You will also learn about the intermediate value theorem and how we can use it to analyze behaviors when we don't know exactly where the zeros of a polynomial are.

This section will give you the algebraic skills to find zeros of polynomials without the aid of a graphing utility. We will use the skills learned here in the coming sections on finding zeros of polynomials and rational functions. If it has been a while, you may need to recall how to use long division to divide integers.

Synthetic division is another algebraic method for finding roots (x-intercepts) of polynomials. Some people prefer this method because it is tidier than long division. With synthetic division, it is important to understand how to interpret your results.

In this section, we will apply polynomial division techniques to analyze and evaluate polynomials. You will be able to evaluate a polynomial function for a given value using the remainder theorem and the factor theorem. These two techniques work well when the roots of a polynomial are integers. We need to use the rational zeros theorem when we have rational roots. This technique also uses polynomial division but will yield zeros that are rational numbers.

After you finish this section, you will be able to find the complex zeros of a polynomial function. This is the last stage in finding zeros of polynomial functions.

In the first section on rational functions, you will learn about their general characteristics and how to use standard notation to describe them. When you are finished, you will be able to use arrow notation to describe long-run behavior given a graph or an equation. You will also be able to use standard notation to describe local behavior.

Domain and range are essential for rational functions since some inputs make them undefined. It is crucial to understand how to define the domain and range of rational functions because it allows you to determine asymptotes and the long-run behavior of rational functions.

In this section, you will use many of the same tools you used to find zeros of polynomials to find the zeros of rational functions. Finding the zeros (intercepts) will help you graph rational functions without a calculator.

This section will dive deeper into the analytic and algebraic tools for finding the three types of asymptotes found in rational functions. Defining asymptotes will help you graph rational functions without a calculator, determine where the function is undefined, and give you a picture of the general behavior of the function.

In our final section on rational functions, we will bring all the skills we have learned together to draw graphs of ration functions without the help of a calculator. We will use asymptotes, intercepts, and general characteristics of the function in question. You will also be able to determine the equation of a rational function given a graph.

In the last section on exponential functions, you will learn how to apply the compound interest formula and explore continuous growth.

This section will define the key characteristics of the graph of an exponential function, including horizontal asymptotes, long-run behavior, and intercepts.

In this section, you will apply what you know about transformations of functions to graphs of exponential functions. You will perform vertical and horizontal shifts, reflections, stretches, and compressions. You will also investigate how these transformations affect the equation, its domain and range, and the end behavior of the function.

Logarithms are the inverses of exponential functions. You will explore the relationship between an exponential and a logarithmic function. You will also explore the basic characteristics of a logarithmic function, including domain, range, and long-run behavior.

In this section on logarithmic functions, you will explore logarithms with base ten and base e and how they are related to their inverse exponential functions.

Now, we will define the domain and range of a logarithmic function given an equation or a graph. We will also construct graphs of logarithmic functions given tables and equations.

Finally, we will transform the graph of logarithmic functions using vertical and horizontal shifts, reflections, and compressions and stretches. Given the graph of a logarithmic function, we will practice defining the equation.

Before diving into solving logarithmic and exponential equations, it is helpful to know the properties of logarithms because they can help you out of tricky situations. In this section, you will learn the algebraic properties of logarithms, including the power, product, and quotient rules.

Finally, we will wrap up the properties of logarithms by learning how to expand and condense logarithms and use the change of base formula.

Getting the variable out of the exponent can be tricky, but you will learn the basics in this section.

The next step in solving exponential equations involves using their inverse, the logarithm. You will work with several different bases and even with exponents that are expressions.

In this unit, you will explore the techniques for solving logarithmic equations. We will begin by using the definition of a logarithm to "undo" it. Then, we will work up to more complex techniques.

Now, we will solve applied problems that involve half-life and the radioactive decay of chemical elements.

Did you know that you can predict the temperature of a cup of hot tea after it sits for 20 minutes? This section will take a deeper look at applications of exponential functions and the behaviors they model in the real world around us. We will explore half-life, radioactive decay, and Newton's law of cooling.

The spread of a virus such as COVID-19 depends on how many people have the virus and how many people are left in the population to which the virus can spread. This behavior can be modeled using a logistic growth model. In this section, you will learn about the different components of a logistic growth model and what behaviors it is used to model.

Now, we will practice building models using datasets. Here, you will see a short video on using an online graphing calculator to do the calculations required. Billions of data points are collected every year in fields from consumer behavior to weather. Fitting this data to a model allows us to explore the behaviors we observe around us meaningfully. The first model you will build is logarithmic.

This section will introduce the basic characteristics of a system of linear equations.

Now that you know some methods for solving linear systems, it is vital to understand the solution. Linear systems have either one solution, no solutions, or infinitely many solutions, which we can determine by analyzing the solution.

We continue our study of systems of linear equations by learning different methods for solving them. You will learn to use graphs, substitution, and the addition method to solve linear systems.

Systems are great for modeling the relationship between behaviors we can observe. In this section, you will explore the profits of a fictional company. You will combine your knowledge of graphing linear functions, solving systems of linear equations, and analyzing the results to understand when a company will break even.

We will further our study of systems of linear equations in this section by adding a variable. A linear function with three variables is similar to one with two, but the algebra used to solve them can get complex quickly. In this section, you will verify whether an ordered triple is a solution to a linear system with three variables and learn how to use elimination to find a solution.

In this section, you will learn how to analyze the solution to a system with three variables.

Now, we will explore graphical and algebraic methods for finding the solution to a system of equations containing combinations of different equations, including linear, quadratic, and circles.

Finally, we will explore non-linear inequalities. You will graph a quadratic inequality and a system of non-linear inequalities. Graphing non-linear systems of inequalities is similar to graphing inequalities, but you will need to take additional steps to determine the region where there are solutions to both inequalities.

First, we will focus on ellipses and the components that make up the equation for an ellipse. You will learn how to write the equation of an ellipse given different components of the ellipse.

This section will focus on graphing ellipses given equations that are either centered at the origin or not centered at the origin.

Hyperbolas can be constructed by intersecting a right circular cone with a plane at an angle where both cone halves intersect. In this section, you will explore the characteristics of hyperbolas and use them to construct hyperbola equations. We will focus on whether or not the hyperbolas are centered at the origin.

This section will focus on graphing hyperbolas given equations that are centered at the origin and those that are not.

Parabolas can be constructed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, a parabola is formed. In this section, you will explore the characteristics of parabolas and use them to construct equations of parabolas. Note that these are not the parabolas we studied before because they are not functions.

This section will focus on graphing parabolas given equations not centered at the origin.

If you plan to learn calculus, this introduction to sequences and series will be very helpful. The first topic you will explore is how to write the terms of a sequence given a formula.

In this section, you will learn what a recursive formula is and apply it to find the terms of a recursively defined sequence.

We will explore two kinds of sequneces in this unit. The first is the arithmetic sequence. In this section, you will learn the characteristics of arithmetic sequences and use a formula to find the terms.

Now, we will learn how to find the terms of an arithmetic sequence given a recursive formula.

We continue with geometric sequences. Now, we'll cover the characteristics and terms of a geometric sequence.

Finally, we will find the terms of a geometric sequence given a recursive formula.

At the last stop on our journey, we will learn the basic properties of an arithmetic series. We will also learn how to use standard notations to express series.

Finally, we will use formulas to find the terms of a geometric series.