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by: Kiana Thompson


Kiana Thompson
College Algebra
John Graef

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About this Document

Here is a study guide that will help you on the upcoming test. It covers all sections that we will be tested on.
College Algebra
John Graef
Study Guide
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This 15 page Study Guide was uploaded by Kiana Thompson on Friday October 9, 2015. The Study Guide belongs to 1130-02 at University of Tennessee - Chattanooga taught by John Graef in Summer 2015. Since its upload, it has received 68 views. For similar materials see College Algebra in Math at University of Tennessee - Chattanooga.


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Date Created: 10/09/15
MATH EXAM 2 STUDY GUIDE 13 Factoring Factoring writing a number or polynomial as a product of its function A FACTOR is a number or algebraic term that Will divide into another number or algebraic term evenly The GREASTEST COMMON FACTOR GCF is the largest factor that two or more numbers or algebraic terms have in common Factoring trinomials With leading coefficient of 1 Factoring trinomials With leading coefficient other than 1 Difference of squares a2 b2 ab a b Sums of squares cannot be factored can only factor out GCF if there is one 14 Rational Expressions polynomial Rational expression W 7 3 x2 2 x 2 y 3 Examples of rational expressions x 5 5x 7 y 46 17 Quadratic Equations What is the difference between solVing a linear equation and solVing a higher degree polynomial equation Standard for of a quadratic equation 0 2 962 0 a at 0 Why can a not equal zero If a 0 What kind of equation is it Tools needed for solving quadratic equations Zero factor property If xy 0 then x 0 or y 0 or both are equal to zero Square root property If b gt 0 then the solutions of x2 b are x 2 E and xZ db biVb2 4ac Quadratic formula If 0 ax2bx 0 then x Z 2 a a at 0 Even if you can factor to solve a quadratic equation you still have the option of using the quadratic formula 34 Quadratic Functions Quadratic functions can be written in the form fx ax2 bx c a 0 Dissecting 1 If a gt 0 positive the parabola opens up If a lt 0 negative the parabola opens down 2 The Vertex the center point is the highest or lowest on the function depending on Whether the parabola opens up or down Finding the vertex if the function is in standard form fx ax2 bx c b b b b Vertex Z f 2 61 ie 36 5 and yf E b 3 Axis of Symmetry x xvalue of vertex ie x Z 4 The yintercept is c 5 The xintercepts are found When y 0 0 ax2 bx c 6 Find the maximum value if it exists or the minimum value if it exists of the function The yvalue of the vertex will be either the maximum or minimum value of the function depending on which direction the parabola opens 7 Find the domain input values and range output values There are three possibilities when finding xintercepts 1 Two xintercepts 2 One xintercept 3 No xintercepts Also the xvalue of the vertex will be directly between the two xintercepts if they exists 35 Applications of Quadratic Functions Use the applications of quadratic functions to help solve the examples 36 Polynomial Functions Polynomial functions of a higher degree are usually not as easy to graph algebraically and may require calculus tools if you want to graph without using a graphing calculator However there are some basic properties that you should know regarding polynomial functions B asic graphs in the form of agt0andnodd alt0andnodd l3 4th 32 241 41 s w an 115 24 32 a1 40 43 4 2 1 5 32 6 IS IS 4 4 5 5 8 E n quotI agt0andneven alt0andneven The graphs of more complicated polynomial functions have similar end behaviors determined by the leading coefficient and the degree of the polynomial but With PEAKS and VALLEYS in the middle The number of peaks and valleys on the graph of a polynomial function of degree n is at most n l The number of Xintercepts on the graph of a polynomial function of degree n is at most n 37 Rational Functions A rational function is a function Whose rule is the RATIO of TWO POLYNOMIALS 3x 14963 9 Examples of rational functions f x 7x8 x 7x1221x49 Important characteristics of rational functions Vertical Asymptotes If a number 6 makes the denominator zero but not the numerator then x c is a vertical asymptote for the graph of the function Horizontal Asymptotes If the graph of a function approaches a horizontal line very closely When x is very large or very small we say that this line is a horizontal asymptote of the graph Finding asymptotes algebraically Vertical asymptotes occur When the denominator is equal to zero as long as the value does not make the numerator equal to zero Horizontal asymptotes can be found by using calculus or by analyzing the degree of the numerator and denominator of the function 1 If the degree of the numerator is equal to the degree of the denominator the horizontal asymptote is the quotient of the leading coefficients 2 If the degree of the numerator is less than the degree of the denominator the horizontal asymptote is y 0 3 If the degree of the numerator is greater than the degree of the denominator there are no horizontal asymptotes Multjnle Choice and Short Answer 1 Find the greatest common factor for the set 30X3 70X6 60X9 A 30X B 10X3 C 5X3 D 2X3 2 Simplify 32y12 72y13 A 22yl27y2 B 22y 13 C 22y 12 227y D 42y 12 3 Simplify 3X54X56 A X543X52 B X543X210X25 C X54X210X28 D X54 4 Factor X2 5X 4 A x2 x2 B x 2 x 2 C x1 x5 D x4 x1 5 Factor X2 2X 3 A x2 x 1 B x1 x3 C x 1x3 D x 2 x1 6 Factor 2X2 9X 4 A 2x 1x 4 7 Factor X2 9 A x1 x9 X2 4 8 Factor A x2 x2 25p2 9 Simplify 35193 10p A 10 5p D 10p 10z5 10 Simplify 20 Z 10 52z1 A 102z1 10z5 13 102z1 Z2 5x6 11 Simplify Z2 4 B x4 x2 B Xl X9 B x 2 x 2 C 2x4 x 1 C X3 X3 C Prime D 2x1 x4 D X3 X3 D x2x 2 z 3z 2 z 3 A z3 z2 B 22 z2 C Z2Z2 D Z2 2 1 12 Simplify Ex g i E 3 A 15 B 2 C 5 1 D 5 32 13 Simplify 7 T 3 27 2 A B C 2 7 D 5 7x14x3 14 Write in lowest terms I T 66 y 66y 3 A 11 B39 11y C x2 2 x 2 3 15 Simplify 77 2 Z i A 7 B 5 C 49 49 D r2 r 2 16 Write in lowest terms 7 3 A mm 2 B o c i 3 3 7 17 Write in lowest terms m2 3 m10 m2 m20 3 7 10m26 A39 m 5m2 m 5m4 1339 m 5m2m4 25m13 m 5m2m4 18 Solve the quadratic equation 222 42 A 222 4z B z 2 2 C z 0 2 D 22 22 19 Solve the quadratic equation X2 6X 9 10 A X 3 B X2 6X 9 C X 3 3 D X3 X3 20 Solve the quadratic equation z2z7 4 SHOW WORK BELOW 21 Solve using the quadratic formula 4k2 2k 1 SHOW WORK BELOW 22 Solve using the quadratic formula X2 2X 3 0 2141 6 A X 1 3 B 2 C X3 Xl D X l 3 Use h l6t2 Vt h V initial velocity at time 0 h height when t 0 t seconds 23 You are standing on a cliff that is 200 feet high How long will it take the rock to reach the ground if you drop it h l6t2 200 SHOW ALL WORK 24 How long will it take the rock to reach the ground if you throw it downward at an initial velocity of 40 feet per second h 161 2 40t 200 SHOW ALL WORK 11 25 How far does the rock fall in 2 seconds if you throw it downward with an initial velocity of 40 feet per second h2 16t2 40t 200 SHOW ALL WORK 26 Find the vertex of fix 2962 296 3 A 4 1 B 1 4 C 3 1 D 1 3 2 27 Find the axis of symmetry f x x 296 3 Ax1 BX4 Cx1 2 28 Find the yintercept f x Z x 2 x 3 Ay03 By13 Cy14 29 Find the xintercepts f x 2962 2x 3 A x3 x1 B X 31 C X 3 1 2 30 Find the maximum or the minimum f x x 2 x 3 2 31 Find the domain and the range fix 2 6 296 3 D y 3 1 Dx23 12 32 Find the maximum height attained by the object h 80t 162 2 33 Find the number of seconds it takes the object to hit the ground h 80t 162 2 34 How much does it cost per box to make 15 boxes 18 boxes 30 boxes Cx x2 40x 405 35 How many boxes should be made in order to keep the cost per box at minimum What is the minimum cost per box Cx x2 40x 405 36 Find the breakeven point Rx 200x x2 Cx 70x 2200 0 S x S 100 A x110 Bx2200 Cx20 Dx20 37 Use the graph above to find how many peaks and valleys there are 38 Use the graph above to find how many X intercepts there are 39 Find the vertical asymptote of f x 2 1 x 40 Find the horizontal asymptote of f x 2 1 x 13 41 Find the vertical asymptote of 42 Find the horizontal asymptote of WNQP PP P KEEPS o OgtgtUWWOWgtWUOgtW ANSWER KEY 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 gtOOUgt 144 feet fallen B A A C 1 4 MAXIMUM D oo oo R 4 00 100 ft 5 Seconds 15 2 a box 18 050 a box 30 350 a box 20 boxes at 5 a piece C 2 peaks and valleys 3 xintercepts x 1 y 0 x 2 2 42y2 15


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