# College Algebra

This course is a standard course in college-level algebra. Topics include (but are not limited to) properties of functions; polynomial, rational, exponential, logarithmic functions and their graphs; and conic sections. Dual enrollment for college credit may be available.

**Credits:** 1.0 (0.5 per semester)

**Estimated Completion Time:** 2 semesters/18-36 weeks

**State Course Number: **115810

Dual Credit

NCAA Approved

**Unit 1: Algebraic Expressions**

**Unit Overview:** In this unit, you will review the key algebraic skills and knowledge necessary for being successful in the course. These are all topics that you should have been exposed to prior to enrolling in the course.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3)3 = 5(1/3)3 to hold, so (5 1/3)3 must equal 5. (N.RN.1)
- Rewrite expressions involving radicals and rational exponents using the properties of exponents. (N.RN.2)
- Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). (A.SSE.2)
- Graph linear and quadratic functions and show intercepts, maxima, and minima. (F.IF.7a)
- Represent and solve equations and inequalities graphically. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (A.REI.10)

**Unit 2: Solving Equations and Inequalities**

**Unit Overview:** In this unit, you will build the foundational skills for higher-level algebra, explore their practical and theoretical applications, and acquaint yourself with appropriate mathematical notation for being successful in Algebra.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real. (N.CN.1)
- Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex Numbers. (N.CN.2)
- Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (N.CN.3)
- Solve quadratic equations with real coefficients that have complex solutions. (N.CN.7)
- Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction,multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. (A.APR.7)
- Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (A.REI.2)
- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (A.REI.3)
- Solve quadratic equations in one variable. (A.REI.4)

- Use the method of completing the square to transform any quadratic equation in x into an equation of the form(x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
- Solve quadratic equations by inspection (e.g., for x2 = 49),taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of theequation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

- Graph linear and quadratic functions and show intercepts, maxima, and minima. (F.IF.7a)
- Represent and solve equations and inequalities graphically. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (A.REI.10)

**Unit 3: Functions and Graphs**

**Unit Overview:** In this unit, you will develop conceptual and practical understandings of functions and the connection between geometric/spatial reasoning and algebraic principles of manipulating and communicating mathematical expressions.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (F.IF.1)
- Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (F.IF.2)
- For a function that models a relationship between two quantities,interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts;intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (F.IF.4)
- Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (F.IF.7b)
- Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (F.IF.9)
- Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x),f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (F.BF.3)
- Find inverse functions. (F.BF.4)
- Verify by composition that one function is the inverse of another. (F.BF.4)
- Read values of an inverse function from a graph or a table,given that the function has an inverse. (F.BF.4)
- Interpret linear models. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. (S.ID.7)

**Unit 4: Polynomials and Rational Functions**

**Unit Overview:** In this unit, you will apply conceptual understanding of equivalent expressions and generalizable rules of lower-degree equations, functions, and real numbers to higher-degree polynomials and complex numbers.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). (N.CN.8)
- Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (N.CN.9)
- Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (A.SSE.3)
- Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (A.APR.3)

**Unit 5: Rational Functions**

**Unit Overview:** In this unit, you will learn to identify the key components necessary to describe, graph, and predict rational functions.

**Missouri Learning Standard:** You will know you have achieved the learning goal when you can:

- Graph linear and quadratic functions and show intercepts,maxima, and minima. (F.IF.7a)
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (F.IF.7c)
- Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (F.IF.7d)
- Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (F.IF.8)

**Unit 6: Exponents and Logarithms (Functions an Equations)**

**Unit Overview: **In this unit, you will explore how to manipulate and solve equations when the variable is an exponent, and the many applications to science and technology that these concepts have

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (F.IF.7)

- Graph exponential and logarithmic functions, showing intercepts and end behavior.

- Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (F.BF.5)

**Unit 7: Systems of Equations and Inequalities**

**Unit Overview: **In this unit, you will build on your skills of solving equations and graphing functions to develop an understanding of problem-solving with multiple equations simultaneously.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Solve systems of linear equations exactly and approximately (e.g.,with graphs), focusing on pairs of linear equations in two variables. (A.REI.6)
- Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. (A.REI.7)

**Unit 8: Matrices**

**Unit Overview: **In this unit, you will be introduced to using matrices to organize large quantities of information, the operations and procedures used to solve problems in this format, and their applications to computer science, chemistry, and economics.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Add, subtract, and multiply matrices of appropriate dimensions. (N.VM.8)
- Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation,but still satisfies the associative and distributive properties. (N.VM.9)
- Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. (N.VM.10)
- Represent a system of linear equations as a single matrix equation in a vector variable. (A.REI.8)
- Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). (A.REI.9)

**Unit 9: Sequences, Series, and Patterns**

**Unit Overview: **In this unit, you will explore how to identify, communicate, and make predictions with mathematical patterns.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.For example, calculate mortgage payments. (A.SSE.4)
- Know and apply the Binomial Theorem for the expansion of (x + y)nin powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) (A.APR.5)
- Write a function that describes a relationship between two quantities. (F.BF.1)

- Determine an explicit expression, a recursive process, or steps for calculation from a context.

- Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. (F.BF.2)

**Unit 10: Conic Sections**

**Unit Overview: **In this unit, you will connect your previous learning of geometric constructs to algebraic notation and manipulation.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Translate between the geometric description and algebraic equation for a conic section. (GPE.A)

This course is available for dual credit through Ozarks Technical Community College. Find out more about dual credit on the institution's website or by contacting their office:

**Ozarks Technical Community College**

OTC Dual Credit

1001 E. Chestnut Expressway,

Springfield, MO 65802

Phone: 417-447-8198

Email: dualcredit@otc.edu