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

Published: 05-16-2022

This book offers 1,001 opportunities to gain confidence in your math skills. Much more than a workbook, this study aid provides pre-calculus problems ranked from easy to advanced, with detailed explanations and step-by-step solutions for each one. Also included with the book is free online access to the problems so you can practice anywhere, anytime.

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

23 results
23 results
Pre-Calculus: 1001 Practice Problems For Dummies Cheat Sheet

Cheat Sheet / Updated 02-24-2022

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. Also, you find a step-by-step description of how to complete the square — most useful when you’re working with conic sections and other equations with specific formats.

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Counting Techniques for Pre-Calculus

Article / Updated 03-26-2016

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. Use the following formulas for these counting techniques:

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Frequently Used Pre-Calculus Formulas

Article / Updated 03-26-2016

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.

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Completing the Square for Pre-Calculus

Article / Updated 03-26-2016

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 quadratic equations ax2 + bx + c = 0 and 2x2 +7x ‒ 15 = 0 by completing the square:

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Algebra Basics Needed for Pre-Calculus

Article / Updated 03-26-2016

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. If x = 3 and y = x, then y = 3. Answer: transitive The transitive property says that if one value is equal to both a second value and a third, then the second and third values are equal to one another. Find the midpoint of the segment between the points (−5, 2) and (7, −8). Answer: (1, ‒3) Use the midpoint formula, and the given points:

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Beyond the Basics with Trig Identities

Article / Updated 03-26-2016

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. These trigonometric identities are broken into groups, depending on whether you’re trying to combine angles or split them apart, increase exponents or reduce them, and so on. The groupings can help you to decide which identity to use in which situation. Keep a list of these identities handy, because you’ll want to refer to them as you work through the problems. You’ll work with the more-advanced trig identities in the following ways: Using the function values of two angles to determine the function value of the sum of the angles Applying the identities for the difference between two angles Making use of the half-angle identities Working from product-to-sum and sum-to-product identities Using the periods of functions in identities Applying power-reducing identities Deciding which identity to use first When you’re working on these particular trig identities, some challenges will include the following: Applying the identities using the correct order of operations Simplifying the radicals correctly in half-angle identities Making the correct choices between positive and negative identities Practice problems Use a sum or difference identity to determine the missing term in the identity: Answer: Use the cosine-of-a-difference identity: Use a double-angle identity to determine the missing term in the identity. Answer: 0 Replace with the double-angle identity involving the cosine: Replace the 1 with from the Pythagorean identity:

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Complex Numbers in Pre-Calculus

Article / Updated 03-26-2016

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. You’ll work on complex numbers in the following ways: Simplifying powers of i into one of four values Adding and subtracting complex numbers by combining like parts Multiplying complex numbers and simplifying resulting powers of i Dividing complex numbers by multiplying by a conjugate When working with complex numbers, some challenges will include the following: Multiplying imaginary numbers correctly Choosing the correct conjugate and simplifying the difference of squares correctly when dividing complex numbers Practice problems Write the power of i in its simplest form: i301 Answer: i Rewrite the exponent as the sum of a multiple of 4 and a number between 0 and 3: i301 = i75(4) + 1 Now write the power of i as the product of two powers: i75(4) x i1 The value of i4n is 1, so i301 = 1 x i1 = i. Multiply. Write your answer in a + bi form: (2 – 3i)(2 + 3i) Answer: 13 Use FOIL to multiply the binomials:

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Conic Sections in Pre-Calculus

Article / Updated 03-26-2016

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. If you come along and slice one of those cones parallel to the ground, the cut edges form a circle. Slice the cone on an angle, and you have an ellipse. Slice the cone parallel to one of the sides, and you have a parabola. And, finally, slice through both cones together, perpendicular to the ground, and you have a hyperbola. If these descriptions just don’t work for you, the practice problems should do the trick. You’ll work on conic sections in the following ways: Recognizing which conic you have from the general equation Finding the centers of circles and ellipses Determining the foci of circles, ellipses, and parabolas Using the directrix of a parabola to complete the sketch Writing the equations of a hyperbola’s asymptotes Changing basic conic section equations from parametric to rectangular When working with conic sections, some challenges will include the following: Determining the major axis of an ellipse Sketching the graph of a parabola in the correct direction Using the asymptotes of a hyperbola correctly in a graph Finding the square root in the equation of a circle when finding the radius Practice problems Name the conic and its center. Answer: ellipse; center: (–4, 1) The standard form for the equation of an ellipse with center (h, k) is The given equation is already in this form, so you can identify the center coordinates by looking at the values substituted for h and k. Write the equation of the circle described. Then graph the circle: center: (4, 3); radius: 5 Answer: (x – 4)2 + (y – 3)2 = 25 The standard equation of a circle with radius (h, k) and radius r is (x –h)2 + (y – k)2 = r2. Substitute the given point (4, 3) for the (h, k) and square the 5: Credit: Illustration by Thomson Digital

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Exponential and Logarithmic Functions Used in Pre-Calculus

Article / Updated 03-26-2016

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. Exponential and logarithmic functions can have bases that are any positive number except the number 1. The special cases are those with base 10 (common logarithms) and base e (natural logarithms), which go along with their exponential counterparts. The whole point of these functions is to tell you how large something is when you use a particular exponent or how big of an exponent you need in order to create a particular number. These functions are heavily used in the sciences and finance, so studying them here can pay off big time in later studies. You’ll work with exponential and logarithmic functions in the following ways: Evaluating exponential and log functions using the function rule Simplifying expressions involving exponential and log functions Solving exponential equations using rules involving exponents Solving logarithmic equations using laws of logarithms Graphing exponential and logarithmic functions for a better view of their powers Applying exponential and logarithmic functions to real-life situations Don’t let common mistakes trip you up. Here are some of the challenges you’ll face when working with exponential and logarithmic functions: Using the rules for exponents in various operations correctly Applying the laws of logarithms to denominators of fractions Remembering the order of operations when simplifying exponential and log expressions Checking for extraneous roots when solving logarithmic equations Practice problems Graph the exponential function: f(x) = –3x Answer: Credit: Illustration by Thomson Digital You find the x-intercepts by solving for f(x) = 0. No values of x make the equation true, so there are no x-intercepts. You find the y-intercept by substituting 0 for x: f(0) = –30 = –1 So the y-intercept is (0, –1). There’s a horizontal asymptote at y = 0 because the limit as x approaches is 0. The function is decreasing as x approaches because the values of the function are getting smaller and smaller, and the function approaches 0 as x approaches because of the horizontal asymptote. Solve the exponential equation for x: Answer: x = ‒5 First, rewrite the right side of the equation so that it has the same base as the left: The bases are now the same, so set the exponents on each side equal to each other: x = –5

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Function Basics for Pre-Calculus

Article / Updated 03-26-2016

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. Functions have some special properties and operations that allow for investigation into what happens when you change the rule. In pre-calculus, you’ll work with functions and function operations in the following ways: Writing and using function notation Determining the domain and range of different types of functions Recognizing even and odd functions Checking on whether a function is one-to-one Finding inverses of one-to-one functions Performing the basic operations on functions and function rules Working with the composition of functions and the difference quotient Don’t let common mistakes trip you up; keep in mind that when working with functions, your challenges will include Following the order of operations when evaluating functions Determining which values need to be excluded from a function’s domain Working with negative signs correctly when checking for even and odd functions Being sure a function is one-to-one before trying to determine an inverse Correctly applying function rules when performing function composition Raising binomials to higher powers and including all the terms Practice problems Find the domain and range for the function. Credit: Illustration by Thomson Digital Answer: domain: –5 < x; range: or y < –1. The domain is the set of x values, and the range is the set of y values for which the function is defined. In this case, x is defined for all real numbers greater than −5. You don’t include −5 because there’s an open dot on −5 in the graph. If then the range is If x > 1, then the range is all real numbers less than −1. So the range is or y < –1. Find the inverse of the function: Answer: Change f(x) to y: Interchange x and y: Now solve for y: Rename y as f–1(x):

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