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Pre-Calculus For Dummies
Author:
Published:
November 13, 2018
Overview
Get ahead in pre-calculus
Pre-calculus courses have become increasingly popular with 35 percent of students in the U.S. taking the course in middle or high school. Often, completion of such a course is a prerequisite for calculus and other upper level mathematics courses.
Pre-Calculus For Dummies is an invaluable resource for students enrolled in pre-calculus courses. By presenting the essential topics in a clear and concise manner, the book helps students improve
their understanding of pre-calculus and become prepared for upper level math courses.
- Provides fundamental information in an approachable manner
- Includes fresh example problems
- Practical explanations mirror today’s teaching methods
- Offers relevant cultural references
Whether used as a classroom aid or as a refresher in preparation for an introductory calculus course, this book is one you’ll want to have on hand to perform your very best.
When you study pre-calculus, you are crossing the bridge from algebra II to Calculus. Pre-calculus involves graphing, dealing with angles and geometric shapes such as circles and triangles, and finding absolute values. You discover new ways to record solutions with interval notation, and you plug trig identities into your equations.
About The Author
Mary Jane Sterling is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and oodles of other Dummies titles. She was a Professor of Mathematics at Bradley University in Peoria, Illinois, for more than 30 years.
Articles From The Book
3 results
Pre-Calculus
Articles
Here you find some pretty amazing curves that are formed from some pretty simple function equations. The trick to drawing these polar curves is to use radian measures for the input variables and put the results into a polar graph. A polar graph uses angles in standard positions and radii of circles; it’s not your usual rectangular coordinate system.
Pre-Calculus
Articles
Circles are simple to work with in pre-calculus. A circle has one center, one radius, and a whole lot of points, but you follow slightly different steps, depending on whether you are graphing a circle centered at the origin or away from the origin.The first thing you need to know in order to graph the equation of a circle is where on a plane the center is located.
Pre-Calculus
Articles
The 30-60-90 triangle is shaped like half of an equilateral triangle, cut straight down the middle along its altitude. It has angles of 30 degrees, 60 degrees, and 90 degrees, thus, its name! In any 30-60-90 triangle, you see the following: The shortest leg is across from the 30-degree angle, the length of the hypotenuse is always double the length of the shortest leg, and you can find the length of the long leg by multiplying the short leg by the square root of 3.
Pre-Calculus
Articles
Almost every function has an inverse. An inverse function basically undoes a function. The trigonometric functions sine, cosine, and tangent all have inverses, and they're often called arcsin, arccos, and arctan.In trig functions, theta is the input, and the output is the ratio of the sides of a triangle. If you're given the ratio of the sides and need to find an angle, you must use the inverse trig function:Here's what an inverse trig function looks like in action.
Pre-Calculus
Articles
A graph for a function that's smooth without any holes, jumps, or asymptotes is called continuous. Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous at some value c in its domain:
f(c) must be defined. The function must exist at an x value (c), which means you can't have a hole in the function (such as a 0 in the denominator).
Pre-Calculus
Articles
As your pre-calculus teacher will tell you, functions that aren't continuous at an x value either have a removable discontinuity (a hole in the graph of the function) or a nonremovable discontinuity (such as a jump or an asymptote in the graph):
If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it.
Pre-Calculus
Articles
In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should get the original polynomial (this is a great way to check yourself on your factoring skills).One set of factors, for example, of 24 is 6 and 4 because 6 times 4 = 24.
Pre-Calculus
Articles
All local maximums and minimums on a function's graph — called local extrema — occur at critical points of the function (where the derivative is zero or undefined). Don’t forget, though, that not all critical points are necessarily local extrema.The first step in finding a function’s local extrema is to find its critical numbers (the x-values of the critical points).
Pre-Calculus
Articles
In pre-calculus, you may need to find the equation of asymptotes to help you sketch the curves of a hyperbola. Because hyperbolas are formed by a curve where the difference of the distances between two points is constant, the curves behave differently than other conic sections. This figure compares the different conic sections.
Pre-Calculus
Articles
If you need to find the limit of a function algebraically, you have four techniques to choose from: plugging in the x value, factoring, rationalizing the numerator, and finding the lowest common denominator.The best place to start is the first technique. You can only use this technique if the function is continuous at the x value at which you are taking the limit.
Pre-Calculus
Articles
The tangent function has a parent graph just like any other function. Using the graph of this function, you can make the same type of transformation that applies to the parent graph of any function. The easiest way to remember how to graph the tangent function is to remember thatsome interesting things happen to tangent's graph.
Pre-Calculus
Articles
If you’re asked to graph the inverse of a function, you can do so by remembering one fact: a function and its inverse are reflected over the line y = x. This line passes through the origin and has a slope of 1.When you’re asked to draw a function and its inverse, you may choose to draw this line in as a dotted line; this way, it acts like a big mirror, and you can literally see the points of the function reflecting over the line to become the inverse function points.
Pre-Calculus
Articles
Each conic section has its own standard form of an equation with x- and y-variables that you can graph on the coordinate plane. You can write the equation of a conic section if you are given key points on the graph.Being able to identify which conic section is which by just the equation is important because sometimes that's all you're given (you won't always be told what type of curve you're graphing).
Pre-Calculus
Articles
Each conic section has its own standard form of an equation with x- and y-variables that you can graph on the coordinate plane. You can write the equation of a conic section if you are given key points on the graph. You can alter the shape of each of these graphs in various ways, but the general graph shapes still remain true to the type of curve that they are.
Pre-Calculus
Articles
Vertical parabolas give an important piece of information: When the parabola opens up, the vertex is the lowest point on the graph — called the minimum, or min. When the parabola opens down, the vertex is the highest point on the graph — called the maximum, or max.Only vertical parabolas can have minimum or maximum values, because horizontal parabolas have no limit on how high or how low they can go.
Pre-Calculus
Articles
Polar coordinates are an extremely useful addition to your mathematics toolkit because they allow you to solve problems that would be extremely ugly if you were to rely on standard x- and y-coordinates. In order to fully grasp how to plot polar coordinates, you need to see what a polar coordinate plane looks like.
Pre-Calculus
Articles
When you’re given a problem in radical form, you may have an easier time if you rewrite it by using rational exponents — exponents that are fractions. You can rewrite every radical as an exponent by using the following property — the top number in the resulting rational exponent tells you the power, and the bottom number tells you the root you’re taking:For example, you can rewriteas
Fractional exponents are roots and nothing else.
Pre-Calculus
Articles
If you have a coefficient tied to a variable on one side of a matrix equation, you can multiply by the coefficient's inverse to make that coefficient go away and leave you with just the variable. For example, if 3x = 12, how would you solve the equation? You'd divide both sides by 3, which is the same thing as multiplying by 1/3, to get x = 4.
Pre-Calculus
Articles
Sometimes you just can’t express both sides of an exponential equation as powers of the same base. When facing that problem, you can make the exponent go away by taking the log of both sides.
When the variable is on one side
For example, suppose you’re asked to solve 43x – 1 = 11. No integer with the power of 4 gives you 11, so you have to use the following technique:
Take the log of both sides.
Pre-Calculus
Articles
Gaussian elimination is probably the best method for solving systems of equations if you don't have a graphing calculator or computer program to help you.The goals of Gaussian elimination are to make the upper-left corner element a 1, use elementary row operations to get 0s in all positions underneath that first 1, get 1s for leading coefficients in every row diagonally from the upper-left to the lower-right corner, and get 0s beneath all leading coefficients.
Pre-Calculus
Articles
In Pre-Calculus, you're going to come across triangles with right angles that vary in degree. This article covers two of the most common right triangles you'll find.
45-45-90 degree triangles
All 45-45-90-degree triangles (also known as 45ers) have sides that are in a unique ratio. The two legs are the exact same length, and the hypotenuse is that length times the square root of 2.
Pre-Calculus
Articles
When you study pre-calculus, you are crossing the bridge from algebra II to Calculus. Pre-calculus involves graphing, dealing with angles and geometric shapes such as circles and triangles, and finding absolute values. You discover new ways to record solutions with interval notation, and you plug trig identities into your equations.
Pre-Calculus
Articles
As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle your next challenge: calculus. In this article, you find ten habits that should be a part of your daily math arsenal. Perhaps you’ve been told to perform some of these tasks since elementary school — such as showing all your work — but other tricks may be new to you.
Pre-Calculus
Articles
Functions can be categorized in many different ways. Here, you see functions in terms of the operations being performed. Here, though, you see classifications that work for all the many types of functions. If you know that a function is even or odd or one-to-one, then you know how the function can be applied and whether it can be used as a model in a particular situation.
Pre-Calculus
Articles
Every good thing must come to an end, and for pre-calculus, the end is actually the beginning — the beginning of calculus. Calculus includes the study of change and rates of change (not to mention a big change for you!). Before calculus, everything was usually static (stationary or motionless), but calculus shows you that things can be different over time.