PRECALCULUS Semester I Exam Review Sheet

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1 PRECALCULUS Semester I Exam Review Sheet Chapter Topic P.1 Real Numbers {1, 2, 3, 4, } Natural (aka Counting) Numbers {0, 1, 2, 3, 4, } Whole Numbers {, -3, -2, -2, 0, 1, 2, 3, } Integers Can be expressed as p/q where p and q are integers (q 0) Rational Numbers Can not be expressed as p/q such as above Irrational Numbers P.2 Exponents and Radicals Properties of Exponents Scientific Notation Addition, Subtraction break from std form to make exponents the same Then add or subtract coefficients, adj to std form. Multiplication, Division multiply coefficients then add/subt exponents Radicals and their Properties Simplifying Radicals Rationalizing Rational Exponents P.3 Polynomials and Factoring Polynomials Definitions Coefficient Degree Leading coefficient Constant term Monomial, binomial, trinomial, polynomial Standard form Polynomial Operations Addition/Subtraction Multiplication General (mult every term in 2 nd poly by every term in 1 st poly) FOIL Special Products Sum and Difference of Same Terms (Conjugates Pairs) (ax + b)(ax b) = a 2 x 2 b 2

2 Square of a Binomial (ax + b) 2 = a 2 x 2 + 2ab x + b 2 (ax - b) 2 = a 2 x 2-2ab x + b 2 Cube of a Binomial u 3 + v 3 = (u + v)(u 2 uv v 2 ) u 3 - v 3 = (u - v)(u 2 + uv v 2 ) Factoring Definitions Prime Irreducible (over reals, over rationals, over integers) Removing Common Factors Factoring Special Polynomial Forms Difference of Two Squares Perfect Square Trinomial Sum or Difference of Two Cubes General Factoring of Trinomials with Binomial Factors Leading Coefficient is 1 Leading Coefficient is not 1 Factoring by Grouping P.4 Rational Expressions Domain of an Algebraic Expression Simplifying Rational Expressions Operations with Rational Expressions Multiplying Rational Expressions Dividing Rational Expressions Adding/Subtracting Rational Expressions Find Common Denominator (LCD?) Examples 6 and 7, Pg. 40,41 Complex Fractions (Pg. 42: Examples 8, 9, 10, 11) Simplifying Factoring Common Quantities with Different Rational Exponents Difference Quotients P.5 Solving Equations Identity an equation that is true for all x Conditional equations an equation that is true for some (or no) x Solving Conditional Equations Linear Equations (Pg. 50: Ex. 1-3) Also, see ex 3 for extraneous solutions Extraneous Solutions may occur when multiplying or dividing by

3 a variable quantity Quadratic Equations Factoring Square Root Principle Completing the Square Quadratic Formula Polynomial Equations of Higher Degree (Some Cases) Factoring Factoring by Grouping Radical Equations (Check for Extraneous Solutions) Involving Rational Exponent Isolate Rational Exponent Term Raise both sides to reciprocal of exponent Involving One Radical Isolate the radical Square both sides Solve for x Verify solutions Involving Two Radicals Isolate one radical Square both sides Isolate 2 nd radical Square both sides again Solve for x Absolute Value Isolate Absolute Value (on left side) Create two non-absolute value equations Set expression inside absolute value equal to the other side of the equation Set expression inside absolute value equal to the opposite of the other side of the equation Solve both equations for x P.6 Solving Inequalities Inequality solutions are sets or intervals and are called the solution set. Solution sets may be Bounded or Unbounded Intervals Properties of Inequalities Two inequalities with the same solution set are equivalent. Properties of Inequalities Transitive a < b and b < c then a < c

4 Addition of Inequalities a < b and c < d then a + c < b + d Addition of a Constant a < b then a + c < b + c Multiplication by a Constant For c > 0, a < b then a c < b c For c < 0, a < b then a c > b c Linear Inequalities Single Inequality Form 3x + 1 > 5x - 4 Double Inequality Form 5 < 4x 3 < 7 Absolute Value Inequalities Isolate Absolute Value quantity on left side If Inequality is Less Than type: Determine if inequality has No Solution or is True For All x If not, then: Set expression inside absolute value < right side Set expression inside absolute value > opposite of left side Solve both of these inequalities to find solution set Other Types of Inequalities Polynomial Inequalities Move all terms to one side of inequality so other side is zero Find the zeros of the polynomial which are called Critical Points n Critical Points divide number line into n+1 intervals Select one convenient Test Point from each Critical Point interval Evaluate polynomial at each Test Point Solution is set of all Critical Point intervals that satisfy inequality Rational Inequalities Find the Domain of a Square Root Function (Square Root Inequality) P.7 Errors and the Algebra of Calculus Algebraic Errors to Avoid (See Examples in Blue Boxes in Text) Errors Involving Parentheses Errors Involving Fractions Errors Involving Exponents Errors Involving Radicals Errors Involving Dividing Out Some Algebra of Calculus (See Examples in Blue Boxes in Text) Unusual Factoring Writing with Negative Exponents Writing a Fraction as a Sum Inserting Factors and Terms

5 P.8 Graphical Representation of Data The Cartesian (x-y) Plane Plotting Points Sketching a Scatter Plot The Distance Formula Finding a Distance Verifying a Right Triangle The Midpoint Formula 1.1 Graphs of Equations Graph Sketching Plug in points x- and y- intercepts symmetry (and symmetry testing) x- axis f(x,-y) = f(x,y) y- axis f(-x,y) = f(x,y) origin f(-x,-y) = f(x,y) Equation of a circle (x h) 2 + (y k) 2 = r 2 where center = (h,k) Standard Form 1.2 Linear Equationss in Two Variables Using Slope Slope-Intercept Form of linear equation y = mx + b Point-Slope Form of linear equation y = m(x x 1 ) +y 1 Slope: Slope of Parallel Lines m parallel = m Slope of Perpendicular Lines m perpendicular = -1/m 1.3 Functions Function vs Relation Functional Relation of x into y: Graphical Form: Vertical Line Test

6 Numeric, Algebraic, Verbal Forms: For every x there is only one y Functional Relation of y into x: For every y there is only one x Graphical Form: Horizontal Line Test Numeric, Algebraic, Verbal Forms: For every y there is only one x Function Notation Dependent variable vs independent variable Given f(x) = 3x + 3, then f(x+ x) = 3(x+ x) + 3 = 3x + 3 x + 3 Piecewise-Defined Functions Step Functions Evaluating Difference Quotients (Pg. 130: Ex. 9) Domains of Functions Domain of a function is all possible x- values Range of a function is all possible y- values Implied Domain is a restricted domain resulting from a practical interpretation of a model (ex. volume would not be negative.) Applications See Pg. 130: Ex.6, 7, Analyzing Graphs of Functions Graphs Finding Domain and Range Vertical Line Test Zeros of a Function Increasing, Decreasing, and Constant Regions Definition of Relative Minimum and Relative Maximum Step Functions Greatest Integer Function: Evaluate Graph using calculator Piecewise-Defined Functions Evaluate Graph using calculator Even and Odd Functions Even: f(-x) = f(x) Odd: f(-x) = -f(x) 1.5 Shifting, Reflecting, and Stretching Graphs Shifting Left and Right Shifts Left c units: f(x) = x 2 g(x) = (x+c) 2 Shifts Right c units: f(x) = x 2 g(x) = (x-c) 2

7 Shifting Up and Down Shifts Up c units: f(x) = x 2 g(x) = x 2 + c Shifts Down c units: f(x) = x 2 g(x) = x 2 - c Reflecting Reflection about the x- axis: g(x) = -f(x) Reflection about the y- axis g(x) = f(-x) Vertical Stretch/Shrink 1.6 Combinations of Functions Arithmetic Combinations of Functions (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) Composition of Functions (f g) = f(g(x)) Domain of a Composite Function Domain of (f g) = f(g(x)) is all values of x that can be plugged into g(x) such that g(x) can be plugged into f(x) Identifying Composite Functions See Example 6 on Page Quadratic Functions

8 2.2 Polynomial Functions of Higher Degree 2.3 Long and Synthetic Division of Polynomials

9 2.4 Complex Numbers

10 2.5 Zeros of Polynomial Functions 2.6 Rational Functions

11 2.7 Partial Fraction Decomposition

12 3.1 Exponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs

13 3.3 Properties of Logarithms 3.4 Exponential and Logarithmic Equations

14 3.5 Partial Fraction Decomposition 4.1 Radian and Degree Measure

15 4.2 Trigonometric Functions: The Unit Circle

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