Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

The basics of pre-calculus consist of reviewing number systems, properties of the number systems, order of operations, notation, and some essential formulas used in coordinate graphs. Vocabulary is important in mathematics because you have to relate a number or process to its exact description. For pre-calculus, you’ll work with simplifying algebraic expressions and writing answers in the following ways: Identifying which are whole numbers, integers, and rational and irrational numbers Applying the commutative, associative, distributive, inverse, and identity properties Computing correctly using the order of operations (parentheses, exponents/powers and roots, multiplication and division, and then addition and subtraction) Graphing inequalities for the full solution Using formulas for slope, distance, and midpoint Applying coordinate system formulas to characterize geometric figures Don’t let common mistakes trip you up; keep in mind that when working with simplifying expressions and communicating answers, your challenges will be Distributing the factor over every term in the parentheses Changing the signs of all the terms when distributing a negative factor Working from left to right when applying operations at the same level Assigning points to the number line in the correct order Placing the change in y over the change in x when using the slope formula Satisfying the correct geometric properties when characterizing figures Practice problems Identify which property of numbers the equation illustrates.

The basic trig identities will get you through most problems and applications involving trigonometry. But if you’re going to broaden your horizons and study more and more mathematics, you’ll find some additional identities crucial to your success. Also, in some of the sciences, especially physics, these specialized identities come up in the most unlikely (and likely) places.

Quadratic equations are written in many different formats, depending on what the current application is. Completing the square is helpful when you’re writing conics in their standard form, and you can use this method to solve for the solutions of a quadratic equation. Here’s how to solve for x in the quadrati

Complex numbers are unreal. Yes, that’s the truth. A complex number has a term with a multiple of i, and i is the imaginary number equal to the square root of –1. Many of the algebraic rules that apply to real numbers also apply to complex numbers, but you have to be careful because many rules are different for these numbers.

Conic sections can be described or illustrated with exactly what their name suggests: cones. Imagine an orange cone in the street, steering you in the right direction. Then picture some clever highway engineer placing one cone on top of the other, tip to tip. That engineer is trying to demonstrate how you can create conic sections.

Counting the number of ways to perform a task is fairly simple — until the number of choices gets too large. Here are three counting techniques used in pre-calculus: Permutations: How many ways you can choose r things from a total of n things when the order of your choices matters Combinations: How many ways you can choose r things from a total of n things when the order of your choices doesn’t matter Multiplication principle: How many different possibilities exist if you choose 1 from a things, 1 from b things, 1 from c things, 1 from d things, and so on.

Exponential and logarithmic functions go together. You wouldn’t think so at first glance, because exponential functions can look like f(x) = 2e3x, and logarithmic (log) functions can look like f(x) = ln(x2 – 3). What joins them together is that exponential functions and log functions are inverses of each other.

Mathematical formulas are equations that are always true. You can use them in algebra, geometry, trigonometry, and many other mathematical applications, including pre-calculus. Refer to these formulas when you need a quick reminder of exactly what those equations are and which measures or inputs are needed.

A function is a special type of rule or relationship. The difference between a function and a relation is that a function has exactly one output value (from the range) for every input value (from the domain). Functions are very useful when you’re describing trends in business, heights of objects shot from a cannon, times required to complete a task, and so on.

You need to become more familiar with the possibilities for rewriting trigonometric expressions. A trig identity is really an equivalent expression or form of a function that you can use in place of the original. The equivalent format may make factoring easier, solving an application possible, and (later) performing an operation in calculus more manageable.

The graphs of trigonometric functions are usually easily recognizable — after you become familiar with the basic graph for each function and the possibilities for transformations of the basic graphs. Trig functions are periodic. That is, they repeat the same function values over and over, so their graphs repeat the same curve over and over.

You can graph functions fairly handily using a graphing calculator, but you’ll be frustrated using this technology if you don’t have a good idea of what you’ll find and where you’ll find it. You need to have a fairly good idea of how high or how low and how far left and right the graph extends. You get information on these aspects of a graph from the intercepts (where the curve crosses the axes), from any asymptotes (in rational functions), and, of course, from the domain and range of the function.

In mathematics, a limit suggests that you’re approaching some value. Some functions, such as a rational function with a horizontal asymptote, have a limit as the x values move toward positive or negative infinity — that is, as the value of x gets very small or very large. Limits are another way of describing the characteristics of particular functions.

You’ll work on graphing complex numbers. Polar coordinates are quite different from the usual (x, y) points on the Cartesian coordinate system. Polar coordinates bring together both angle measures and distances, all in one neat package. With the polar coordinate system, you can graph curves that resemble flowers and hearts and other elegant shapes.

Polynomial functions have graphs that are smooth curves. They go from negative infinity to positive infinity in a nice, flowing fashion with no abrupt changes of direction. Pieces of polynomial functions are helpful when modeling physical situations, such as the height of a rocket shot in the air or the time a person takes to swim a lap depending on his or her age.

Pre-calculus draws from algebra, geometry, and trigonometry and combines these topics to prepare you for the techniques you need to succeed in calculus.This cheat sheet provides the most frequently used formulas, with brief descriptions of what the letters and symbols represent. Counting techniques are also here, letting you count numbers of events without actually having to list all the ways to do them.

The object of solving equations and inequalities is to discover which number or numbers will create a true statement in the given expression. The main techniques you use to find such solutions include factoring, applying the multiplication property of zero, creating sign lines, finding common denominators, and squaring both sides of an equation.

A system of equations is a collection of two or more equations involving two or more variables. If the number of equations is equal to the number of different variables, then you may be able to find a unique solution that’s common to all the equations. Having the correct number of variables isn’t a guarantee that you’ll have that solution, and it’s not terrible if a unique solution doesn’t exist; sometimes you just write a rule to represent the many solutions shared by the equations in the collection.

A system of inequalities has an infinite number of solutions (unless it has none at all). You solve these systems using graphs of the separate statements. The techniques for solving these systems involve algebraic manipulation and/or matrices and matrix mathematics. When you’re solving your own systems, the approach you use is up to you.

You may think that you’re “not in Kansas anymore” when you leave the familiar world of right triangles and Mr. Pythagoras to enter this new world of oblique triangles. Trigonometry allows for some calculations that aren’t possible with the geometric formulas and other types of measurement. The Law of Sines and Law of Cosines are relationships between the sides and angles of triangles that aren’t right triangles.

Trigonometric functions are special in several ways. The first characteristic that separates them from all the other types of functions is that input values are always angle measures. You input an angle measure, and the output is some real number. The angle measures can be in degrees or radians — a degree being one-360th of a slice of a circle, and a radian being about one-sixth of a circle.

A sequence is a list of items; in mathematics, a sequence usually consists of numbers such as 1, 2, 4, 8, . . . or 1, 1, 2, 3, 5, 8, 13, . . . See the patterns in these two sequences? You need to know how to find patterns in lists of numbers so you can write the rest of the numbers in the list. You can classify many sequences by type, which helps you determine particular terms or sums.

A series is the sum of a list of numbers, such as 1 + 2 + 4 + 8. Many times, you can find a formula to help you add up the numbers in a series. Formulas are especially helpful when you have a lot of numbers to add or if they’re fractions or alternating negative and positive. You can classify many series by type, which helps you determine particular terms or sums.