Review of Intermediate Algebra Content


 Lesley Kennedy
 2 years ago
 Views:
Transcription
1 Review of Intermediate Algebra Content
2
3 Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6 Factoring the Difference of Two Squares... 9 Factoring the Sum and Difference of Two Cubes... Factoring Strategy... Quadratic Equations... 8 Applications of Quadratic Equations... 0 Rational Epressions... Comple Fractions... 8 Rational Equations... 0 Applications of Rational Equations... Linear Equations... 0 Functions... 6 Algebra and Composition of Functions... 5 Eponents... 5 Radicals Rational Eponents... 6 Radical Equations Comple Numbers Solving Quadratic Equations... 7 Solving Quadratic and Rational Inequalities... 8 Equations That Can Be Written in Quadratic Form Graphing a Quadratic Function of the Form f ( ) a b c( a 0) = Pythagorean Theorem... 9 Eponential and Logarithmic Functions Vocabulary Used in Application Problems...00 Review of Intermediate Algebra...0 i Revised: May 0
4
5 FACTORING FACTORING OUT THE GREATEST COMMON FACTOR. Identify the TERMS of the polynomial.. Factor each term to its prime factors.. Look for common factors in all terms.. Factor these factors out. 5. Check by multiplying. Eamples: Factor y Terms: 6, 8, and y + y y Factor into primes + y y Look for common factors Answer: ( y ) Check: ( ) ( y ) ( ) + Factor out the common factor + Check by multiplying y FACTORING TRINOMIALS OF THE FORM: + b + c To factor , look for two numbers whose product = 6, and whose sum = 5. Since the numbers and work for both, we will use them. = 6 + = 5 So, = ( + )( + ) Answer: ( + )( + ) Check by multiplying: ( )( ) List factors of 6: 6 Choose the pair which 6 adds to 5 = 6 + = =
6 FACTORING FACTORING OUT GCF AND FACTORING TRINOMIALS OF THE FORM: +b+c Problems Answers Factor ( 5 + 6). 7 5 y y y + 7y ( y + y ). 6 ( 9)( + 7). + ( 6)( ) 0 8
7 FACTORING FACTORING TRINOMIALS OF THE FORM: Eample: Factor STEP. Factor out the GCF ( ) a +b+c NOTES GCF is.. Identify the a, b, and c a = ; b= 7; and c= for the epression Find the product of ac ac = = 6. List all possible factor pairs that equal ac and identify the pair whose sum is b = 7 + = 5 and 6 and and are the only factor pairs, however and 6 is the correct pair. 5. Replace the b term with the boed numbers You have replaced + 7 with Group the two pairs. ( + ) + ( 6+ ) 7. Factor the GCF out of each pair. ( + ) + ( + ) The resulting common factor +. is ( ) 8. Factor out ( + ) ( )( ) + + These are the two factors of Recall the original GCF from Step. ( ) = ( + )( + ) There are factors of Answer: ( +)( +) 0. Check by multiplying. ( )( ) + + = 6 + +
8 FACTORING FACTORING TRINOMIALS OF THE FORM: Eample: Factor a a a +b+c Hint: Recall the rule of signs: When ac is negative, the factor pairs have opposite signs. STEP NOTES. Factor out the GCF. None. Identify the a, b, and c a = ; b= ; c=. Find the product of ac ac = ( ) =. List all possible factor pairs that equal ac and identify the pair whose sum is b. + ( ) + + ( ) = ( + ) + ( + ) =+ + ( 6) + + ( 6) = ( + 6) = + ( + 6) =+ + ( ) + + ( ) = ( + ) + ( + ) =+ Include all possible factor combinations, including the +/ combinations since ac is negative. 5. Replace the b term with the boed numbers. = + Note: a= a 6a a a a a 6a 6. Group the two pairs. = a + a 6a Group by pairs. The resulting common factor a + is ( ) 7. Factor the GCF out of each pair. ( ) ( ) = a a+ a+ 8. Factor out ( a + ) ( a )( a ) = + These are the factors. Answer: ( a+ )( a ) 9. Check to verify. ( )( ) a+ a = a a
9 FACTORING FACTORING TRINOMIAL PROBLEMS OF THE FORM: a + b + c Problems Answers Factor. ( a+ )( a ) 6a 5a ( a )( a+ ) 0a 5a 0. + ( )( 5 6) ( + 7)( 5+ ) 5
10 FACTORING FACTORING PERFECT SQUARE TRINOMIALS Eample: Factor STEP NOTES. Is there a GCF? No. Determine if the trinomial is a Perfect Square. Is the first term a perfect square? Yes ( ) = Is the third term a perfect square? +8 Yes 9 9 = ( 9) = 8 Is the second term twice the product of the square roots of the first and third terms? +8 Yes ( ) 9 = 8. To factor a Perfect Square Trinomial: Find the square root of the first term. = Identify the sign of the second term + Find the square root of the third term. 9 8 = 9 Write the factored form. Answer: ( +9 ) ( )( ) ( ) = = + 9 Check your work. ( ) ( ) =
11 Eample: Factor 5 + 0y + y STEP NOTES. Is there a GCF? No. Is the trinomial a Perfect Square Trinomial? Is the first term a perfect square? 5 Yes, = 5 = ( ) Is the third term a perfect square? y Yes, y y y = y = ( ) Is the second term twice the product of the square roots of the first and third terms? 0y Yes, ( ) 5 y = 0y. Factor the trinomial. Square root of the first term 5 = 5 Sign of the second term + 0y + Square root of the third term y = y Factored form. ( )( ) ( ) = = 5 + y y y y y Answer: ( 5+ y) Check your work. ( ) ( ) 5+ y 5+ y = 5 + 0y+ y 7
12 FACTORING FACTORING PERFECT SQUARE TRINOMIALS Problems Answers Factor ( ) +. y 8y 9 + ( ) y y 5y + ( ) 5y. 9 0y 5y + + ( ) + 5y 8
13 FACTORING FACTORING THE DIFFERENCE OF TWO SQUARES Eample: Factor: 6 STEP NOTES. Is there a GCF? No. Determine if the binomial is the Difference of Two Squares. Can the binomial be written as a 6, yes difference? Is the first term a perfect square? Yes, = Is the second term a perfect 6 Yes, 6 6 = 6 square?. To factor the Difference of Two Squares Find the square root of the first term Find the square root of the second term Write the factored form. Answer: ( + 6)( 6) = 6 6 = 6 ( )( ) Check your work. ( )( ) 6 = The factors are the product of the sum (+) and difference ( ) of the terms square roots = 6 Eample: Factor 6a b STEP NOTES. Is there a GCF? No. Is the binomial the Difference of Two Squares? Can the binomial be written as a 6a b Yes difference? Is the first term a perfect square? 6a Yes, 6a 6a= 6a Is the second term a perfect b Yes, b b= b square?. To factor the Difference of Two Squares Find the square root of the first 6a 6a = 6a term Find the square root of the b b = b second term Write the factored form. 6a b = Answer: ( 6a+ b)( 6a b) ( 6a+ b)( 6a b) Check your work. ( )( ) 6a+ b 6a b = 6a b The factors are the sum (+) and difference ( ) of the terms square roots. 9
14 FACTORING FACTORING DIFFERENCE OF TWO SQUARES Problems Answers Factor. 9 9 y ( y+ 7)( y 7). 5 6 ( 5 )( 5+ ). ( y)( + y) 9y. 8 6 a ( 9a + )( a+ )( a ) 0
15 FACTORING FACTORING THE SUM OF TWO CUBES Eample: Factor y 5 STEP. Is there a GCF? y = ( 8 + 5y ) NOTES Yes, is the GCF for both terms.. Determine if the binomial is the Sum or Difference of Two Cubes y Remember you are only looking at 8 + 5y ; Sum. Is the first term a perfect cube? 8 Yes, ( ) = = 8 Is the second term a perfect cube? + 5y Yes, 5y 5y 5y = 5y = 5y ( ). To factor the Sum of Two Cubes, use ( )( ) a + b y = a + by a aby + b y where a and b are constants. To factor 8 + 5y : Find the cube root of the first term. 8 = Find the cube root of the second term. 5y 5y = 5y Using the formula above, write the factored form. ( )( ) 8 + 5y = + 5y 0y + 5y Note the sign of the 0y is opposite that of the + 5y. Recall, you factored out the GCF in the final factored form y = ( + 5y)( 0y + 5y ) Answer: ( + 5 y)( 0y + 5 y ) from the original binomial. You must now include it
16 FACTORING FACTORING THE DIFFERENCE OF TWO CUBES Eample: Factor 6 7 y STEP NOTES. Is there a GCF? 6 7 y No. Determine if the binomial is the Sum or Difference of Two Cubes. 6 7 y Difference Is the first term a perfect cube? 6 Yes, ( ) = = 6 Is the second term a perfect cube? 7y Yes, ( ) y y y = y = 7y. To factor the Difference of Two Cubes, use ( )( ) a b y = a by a + aby + b y Where a and b are constants. To factor 6 7 y : Find the cube root of the first term. 6 = Find the cube root of the second term. y 7y = y Using the formula above, write the factored form. ( )( ) 6 7 y = y 6 + y + 9y Note the sign of y is opposite that of the 7y.
17 FACTORING FACTORING THE SUM and DIFFERENCE OF TWO CUBES Problems Answers Factor ( + )( 9 6+ ). 6a 5b ( a 5b)( 6a + 0ab + 5b ). y y y ( y )( y + y+ ) 5
18 FACTORING FACTORING STRATEGY. Is there a common factor? If so, factor out the GCF, or the opposite of the GCF so that the leading coefficient is positive.. How many terms does the polynomial have? If it has two terms, look for the following problem types: a. The difference of two squares b. The sum of two cubes c. The difference of two cubes If it has three terms, look for the following problem types: a. A perfectsquare trinomial b. If the trinomial is not a perfect square, use the grouping method. If it has four or more terms, try to factor by grouping.. Can any factors be factored further? If so, factor them completely.. Does the factorization check? Check by multiplying.
19 FACTORING STRATEGY Always check for Greatest Common Factor first! Then continue: Terms Terms Terms Difference of Two Squares ( ) ( ) + ( )( ) Sum and Difference of Two Cubes 8= + + ( )( ) ( )( ) + 8= + + Perfect Square Trinomials ( ) + ( )( 5) + ( 5) ( + 5)( + 5) ( + 5) Form: + b + c 6+ 5 List factors of Choose the pair which adds to 6 5+ = ( 5) ( 5) ( 5)( ) Form: a + b + c = 0 List factors of Choose the pair which add to = 7 Substitute this pair in for the middle term Factor out the GCF of each pair of terms ( 5 + ) ( 5 + ) ( 5+ )( ) Factor by Grouping ( + ) ( + ) 7 6 ( + )( 7 6) 5
20 FACTORING Problems Factor a p 6q
21 FACTORING Answers. ( + )( )( + ). ( )( + ). ( + )( )( + ). ( + )( ). ( + )( )( ). ( + )( ). ( + )( )( + ) 5. 5 ( + )( 5 ) 5. ( + )( )( 5) 6. ( + 9)( + )( ) 6. ( 5)( 5)( ) + 7. ( + )( 9 6+ ) 7. ( )( 5) + 8. ( a)( + a+ 6a ) 8. ( )( ) + 9. ( 5 p q)( 5 p + 0 pq + 6q ) 9. ( )( ) 0. 0( )( + + ) 0. ( 6 5)( ) +. ( + 5)( 0+ 5). ( )( ) +. ( )( + + 9) 7
22 QUADRATIC EQUATIONS SOLVING QUADRATIC EQUATIONS BY FACTORING. Write the equation in standard form: a + b + c = 0 or 0 = a + b + c.. Factor completely.. Use the zerofactor property to set each factor equal to 0.. Solve each resulting linear equation. 5. Check the results in the original equation. Eample: Solve 00 = 0 This equation is in standard from. ( )( ) = 0 Factor completely. 0 = = 0 Set each factor equal to 0. = 0 = 0 Solve each linear equation. Check: = 0 Check: = 0 00 = 0 00 = 0 ( ) 0 00 = 0 ( ) 0 00 = = = 0 0 = 0 True 0 = 0 True The solutions of 00 = 0 are 0 = and = 0. Eample: Solve 6 = 7+ This equation is in standard 6 7 = 0 form. ( + )( ) = 0 Factor completely. + = 0 = 0 Set each factor equal to 0. Solve each linear equation. = = Check: = Check: = 6 = 7+ 6 = = 7 + = = + 6 = + 9 = True 7 7 = True The solutions of 6 7 = + are = and =. 8
23 QUADRATIC EQUATIONS Problems Answers Solve = 0 = 7, = = 5 = =. 5 = t 5 t = 5 t =. 8y 6y = 0 y = 0 y = 5. ( 0) 8 + = = = = 0 = 5 7. ( 6) = + = 0, =, = 9
24 APPLICATION OF QUADRATIC EQUATIONS Eample: Sharon and Diane went to the deli department at a local grocery store. Sharon arrived first and took a number to reserve her turn for service. Diane took the net number. If the product of their two numbers was 0, what were their numbers? Define the variable. Write and equation ( ) Solve the equation n = Sharon s number n + = Diane s number n n+ = 0 n + n= 0 n + n 0 = 0 ( n+ )( n 0) = 0 n = n = 0  is an etraneous solution Sharon s number: n = 0, Diane s number: n + = Check in the original problem. The answers are: Sharon took number 0 and Diane took number. Eample: Anita s hat blows off her head at the top of an amusement park ride. The height h, in feet, of the hat t seconds after it blew off her head is given by h= 6t + 6t+. After how many seconds will the hat hit the ground? Define the variables h=height of the hat (in feet) t =time (in seconds) Use the given equation h= 6t + 6t+. Substitute h = 0 because the height of the hat is 0 feet when it hits the ground. Solve the Equation 0 = 6t + 6t+ 0 = 6( t t ) 0 = 6( t )( t + ) t = 0 t+ = 0 t = t = is an etraneous solution Check in the original problem. The answer is: It took seconds for the hat to hit the ground. 0
25 APPLICATION OF QUADRATIC EQUATIONS Problems Answers Solve. Bob has two routes he could take to get from his house to the mall. Bob s House a) 8 miles b) 6 miles 7 Mall 5 One route is direct and along a country road for 7 miles. The other route is miles south and 5 miles due east to the mall. a) Find the distance. b) Determine how many miles he saves driving the direct route.. Phil teaches in two different rooms. The room numbers are consecutive even integers whose product is 68. Find the two room numbers. and. The area of a garden is 95ft. The length is feet more than twice the width. Find the dimensions of the garden. 5 ft. and ft.. Suppose that a cannon ball is shot upward with an initial speed of 80 ft./second. The height h in feet of the cannon ball t seconds after being shot is given by h= 6t + 80t. After how many seconds will it hit the ground? 5 seconds
26 RATIONAL EXPRESSIONS SIMPLIFYING RATIONAL EXPRESSIONS Eample: Simplify 6 To simplify, factor numerator and denominator completely and cancel common factors. Recall =. 6 ( + )( ) ( ) + = = Eample: Simplify Recall: Step of factoring strategy: The leading coefficient must be positive. Continue by factoring the numerator and denominator completely. Cancel common factors ( ) = = = ( )( ) ( + )( + 5) ( ) + 5 OPERATIONS WITH RATIONAL EXPRESSIONS To add, subtract, multiply or divide rational epressions, use the following rules for adding, subtracting, multiplying or dividing fractions. Recall:. Multiply two fractions by multiplying the numerators and multiplying the denominators. Use cancellation, if necessary, to write the product in lowest terms. Eample: Simplify ( + )( ) ( + ) ( + )( ) = = + +
27 RATIONAL EXPRESSIONS. Divide two fractions by inverting the second fraction (divisor) and multiplying. Eample: Simplify a a a + a a+ a a a+ a a+ = a a+ a a a+ ( a+ )( a ) a + ( a ) ( a )( a ) ( a + ) ( a ) = =. To add (or subtract) like fractions, add the numerators only and place your result over the common denominator. Eample: Simplify = = To add (or subtract) unlike fractions, find the lowest common denominator, rewrite fractions with the common denominator, and add (or subtract) numerators, placing the answer over the common denominator. Eample: Simplify + + ( )( ) ( ) ( )( ) + = = + ( + )( ) ( + )( ) + = = ( + )( ) ( + )( ) Lowest common denominator is +. which is ( )( ) IMPORTANT NOTE: All answers must be simplified, i.e. reduced to lowest terms.
28 RATIONAL EXPRESSIONS Eamples:. Multiply: + + ( + )( ) = + ( + )( ) ( + )( ) = ( )( ) ( ). Multiply: 5 a a a 5 a = a a a a ( )( ) ( ) a 5 a a + a+ a = ( a + a+ ) a = a + ( + )( ) a a a a a a + + = a +. Divide: = ( + )( ) + 7 = ( ) = ( )( ) ( )( ). Subtract like fractions: ( ) ( + ) + 5 = = ( + ) + 9 = + = = ( + ) ( + )
29 RATIONAL EXPRESSIONS 5. Subtract unlike fractions: 5 y 7 5 y 7 = y y y y ( ) = 5 y 7 + y y = 5 + y 7 y 6. Subtract unlike fractions: y+ y+ y+ y+ = ( )( ) ( )( ) y y y y y y y y ( )( ) ( )( )( ) ( y+ )( y+ ) ( y+ )( y ) ( y 6)( y )( y+ ) ( ) y + y+ y + y ( y 6)( y )( y+ ) ( )( ) ( )( )( ) y+ y+ y+ y = y 6 y y+ y 6 y+ y = = y + y+ y y+ y + y+ = = ( y 6)( y )( y+ ) ( y 6)( y )( y+ ) 5
30 RATIONAL EXPRESSIONS Problems Answers Perform any operations. Simplify ( ) ( + ) y + y y y ( )( ) ( 5)( + ) ( + )( ) ( + ) + ( ) ( ) ( + ) ( )
31 RATIONAL EXPRESSIONS Problems Answers ( + ) ( + )( ). + 0 ( )( ) ( + )( ) y y + + 6( + y) ( )( + ) + + Note: To simplify rational epressions, factor the numerator and the denominator completely. Then cancel common factors to write the rational epression in lowest terms. 7
32 COMPLEX FRACTIONS SIMPLIFYING COMPLEX FRACTIONS. If needed, add or subtract in the numerator and/or denominator so that the numerator is a single fraction and the denominator is a single fraction.. Perform the indicated division by multiplying the numerator of the comple fraction by the reciprocal of the denominator.. Simplify the result, if possible. Eample: Simplify + + = + The LCD of the numerator of the comple fraction is. The LCD of the denominator of the comple fraction is. = + = + Simplify the numerator. Simplify the denominator. + = + = = ( + ) ( ) Change division to multiplication by the reciprocal. Cancel common factors. 8
33 COMPLEX FRACTIONS Problems Answers Simplify y y + 6 ( y ) y( y+ ) 5. 5 y y 5 y y+ 5 or y y. 8 b + 5 b 0 bb+0 ( ) 9
34 STEPS TO SOLVE RATIONAL EQUATIONS. Determine which numbers cannot be solutions of the equation.. Multiply all terms in the equation by the LCD of all rational epressions in the equation. This clears the equation of fractions.. Solve the resulting equation. RATIONAL EQUATIONS. Check all possible solutions in the original equation. Eample: Solve y + = y y STEP. y 0 y. y ( y ) + ( y ) = ( y ) y y. ( ) + ( y )( y ) = ( ) + y y+ = y y = 0 ( y )( y ) + = 0 y = 0 y+ = 0 y = y = 6 NOTES The denominator equals 0 when y =. Therefore, y = cannot be a solution. Multiply all terms in the equation by the LCD of ( y ). Cancel common factors. This clears the equation of fractions. Solve the equation. Check y = and y = Check all possible solutions in the original equation. Check y = Check y = + = + = = + = + = + = = The solutions are y = and y =. 0
35 RATIONAL EQUATIONS Eample: Solve 5 y + + = y y STEP. y 0 y NOTES The denominator equals 0 when y =. Therefore, y = cannot be a solution.. 5 y + y y ( y )( ) + ( y ) = ( y ) Multiply all terms in the equation by the LCD of y. Cancel common factors.. ( y )( ) + 5= y+ This clears the equation of fractions. y 8+ 5= y+ y = y+ y = Solve the equation.. Check y = = = 0 0 Check all possible solutions in the original equation. Cannot divide by 0. See Step. y = cannot be a solution. No solution ( is etraneous)
36 RATIONAL EQUATIONS Eample: Solve 6 = STEP NOTES The denominator equals 0 when = and = 7. Therefore = and = 7 cannot be solutions.. ( + 7) = 6( + ) This eample is a proportion so we can use the cross product.. + 7= = 0 + = 0 = = ( )( ) Solve this equation.. Remember to Check. The solutions are = and =.
37 RATIONAL EQUATIONS Problems Answers Solve. = y y y = 5. + = + 8 =. y 6 = y+ y+ 7 y =, y =. y 5 = y 5 y 5 No solution (5 is etraneous) = 9 + = 0
38 APPLICATIONS OF RATIONAL EQUATIONS NUMBER PROBLEM Eample: If the same number is added to both the numerator and denominator of the fraction 7, the result is. Find the number. Let = the unknown number + = 7+ + = 7+ ( + ) = 7 ( + ) 6 + = = = 5 Define the variable. Form an equation. Solve the equation. Check: + 5= = 9 = = Check the result in the original word problem. Answer: The number is 5. State the conclusion.
39 APPLICATIONS OF RATIONAL EQUATIONS UNIFORM MOTION PROBLEM d d Distance = Rate Time d = r t r = t = t r Eample: Tammi bicycles at a rate of 6 mph faster than she walks. In the same time it takes her to bicycle 5 miles she can walk miles. How fast does she walk? Rate Time = Distance bicycling w w + 6 walking w w Define the variable: w = Tammi s walking rate Check: 5 = w+ 6 w ( w ) 5w= + 6 5w= w+ w = w = walking rate is mph ( w = ) bicycling rate is 0 mph ( w + 6 = 0) Form an equation which sets the times equal to each other. (bicycling time = walking time) Solve the equation. Check the result in the original word problem. Using d t =, show that the walking time equals the bicycling time r t = walking miles mph bicycling 5miles t = 0mph t = hour t = hour The times are the same. Answer: Tammi s walking rate is mph. State the conclustion. 5
40 APPLICATIONS OF RATIONAL EQUATIONS WORK SHARED PROBLEM Work Completed = Work Rate Time Worked w= r t Work Completed by Unit + Work Completed by Unit = Job Completed Eample: It takes Charlie 6 hours to prepare the soccer field for a game. It takes Scotty 8 hours to prepare the same field. How long will it take if they work together? Work rate Time worked = Work completed Charlie t t 6 6 Scotty t t 8 8 Define the variable: t = time in hours that they worked together. Work Completed + Work Completed = Job Completed by Charlie by Scotty t t + = Form an equation. 6 8 t t + = 6 8 t + t = 7t = t = = hours 7 7 Solve the equation. Check: Using w= r t Charlie s work completed w = 6 7 Scotty s work completed w = 8 7 Check the result in the original word problem. w = of the job 7 + = job completed 7 7 w = of the job 7 Answer: It took them 7 hrs. working together. State the conclusion. 6
41 APPLICATIONS OF RATIONAL EQUATIONS SIMPLE INTEREST PROBLEM Interest = Principal Rate Time I I I = p r t = p = t rt pr Eample: An amount of money can be invested in a mutual fund which would yield $50 in one year. The same amount of money invested in a bond would yield only $50 in a year, because the interest paid is % less than that paid by the mutual fund. Find the rate of interest paid by each investment. Mutual Fund Bond Principal Rate Time = Interest 50 r 50 r 50 r.0 50 r.0 ( ) Define the variable: r = rate of interest paid by mutual fund r = Form an equation r.0 which sets the principals equal to each other. 50r 7 = 50r 00r = 7 Solve the equation. r =.05 =.5% r.0 =.05=.5% Check: Find the principal for each investment using I P = rt Check the result in the original word problem. Mutual Fund 50 P = = $0, 000 (.05)( ) Bond 50 P = = $0, 000 (.05)( ) The principals are the same. Answer: The interest rate for the mutual fund is.5% The interest rate for the bond is.5% State the conclusion 7
42 APPLICATIONS OF RATIONAL EQUATIONS Problems. If the same number is subtracted from both the numerator and the denominator of 0 9, the result is. Find the 5 number. Answers. If the numerator of 5 6 is increased by a certain number and the denominator is doubled, the result is. Find the number.. Melissa can drive 0 miles in the same time as it takes her to bike 0 miles. If she can drive 0 mph faster than she can bike, how fast can she bike? 5 mph. Mike can bicycle 6 miles in the same time that he can jog 9 miles. If he can bike 9 mph faster than he can jog, how fast can he bike? mph 8
43 APPLICATIONS OF RATIONAL EQUATIONS Problems 5. In year, an investor earned $60 interest on money he deposited in his savings account at a bank. He later learned that the money would have earned $50 had he invested in municipal bonds, because the bonds paid % more interest at the time. Find the rate he received in his savings account. Also, find the principal amount he put in the savings account. Answers % and $ A plane flies 780 miles downwind in the same amount of time as it takes to travel 580 upwind. If the plane can fly 0 mph in still air, find the velocity of the wind. 50 mph 7. Two bond funds pay interest rates that differ by %. Money invested for year in the first fund earns $0 interest. The same amount invested in the second fund earns $60. Find both the lower rate of interest and the principal amount of the investment. % and $ If the manager of a pet store takes 8 hours to clean all of the cages and it takes his assistant hours more than that to clean the same cages, how long will it take if they work together? 9 hours 9. If it takes a night security guard to make his rounds on campus hours and it takes his assistant hours, how long will it take them working together? 5 hours 9
44 LINEAR EQUATIONS The standard form of the equation of a line is written as a + by + c = 0 where a and b are not both 0. Definition of slope: Let L be a line passing through the points ( y ) and ( y ) of L is given by the formula:. Then the slope y y m = The slopeintercept form of the equation of a line is written as y = m + b where m is the slope and the point ( 0,b) is the yintercept. The pointslope form of the equation of a line passing through (, y ) written as ( ) y y = m and having slope m is where m is the slope and (, y ) is the point. Parallel Property: Parallel lines have the same slope. Perpendicular Property: When two lines are perpendicular, their slopes are opposite reciprocals of one another. The product of their slopes is. 0
45 LINEAR EQUATIONS Eample: Find the equation of a line passing through ( 6,0 ) and ( 8, ) Find the slope of the line through the points. y y m =. 0 m = m = m = Use the pointslope form to find the equation of a line with Using ( 6,0) and m =, substitute into ( ) y y = m y 0 = 6 ( ) y 0 = m = and using one of the points. Answer: y = + 7 This is the equation of the line in slopeintercept form.
46 Eample: Find the equation of a line through ( ) Find the slope of the line. LINEAR EQUATIONS 9, and parallel to y =. y = y = m = This equation is in standard form. Write the equation in slopeintercept form. A line parallel to this line will have the same slope, m = Use m = and point (9, ) to write the equation of the line. ( ) y y = m y = 9 Answer: y = ( ) This is the pointslope form of a line. This is the slopeintercept form of a line. Eample: Find the equation of a line through (9, ) and perpendicular to y =. From the above eample, the line y = has A line perpendicular to this line will have a slope of Use m =. m = and point (9, ) to write the equation of the line. y = 9 ( ) m = (opposite reciprocal). Answer: y = +
47 LINEAR EQUATIONS Problems Answers. Put the following in slopeintercept form. a. 8 y = a. y = b. + y = b. y = +. Find the slope. a. (, ) and ( 7,9 ) a. 5 b. (,) and (, ) b. 5. Find the equation of the line through ( ) 5, and parallel to y = + y = +. Find the equation of the line, and perpendicular to y =. through ( ) y =
48 LINEAR EQUATIONS Find the equation of the line that satisfies the given conditions. Write the equation in both standard form and slopeintercept form. Answers Standard Form SlopeIntercept Form 5. slope = line passes through ( ), y = 5 5 y = + 6. slope = 5 line passes through ( ) 6 0,0 5+ 6y = 0 5 y = 6 7. slope = yintercept y = y = 8. horizontal line through (, ) y = y = slope is undefined and passing through ( 5,6) = 5 cannot be written in the slopeintercept form. 0. slope = intercept 5 + y = 5 5 y =. horizontal line through ( 5, ) y = y = 0
49 LINEAR EQUATIONS Find the equation of the line that satisfies the given conditions. Write the equation in both standard form and slopeintercept form. Answers Standard Form SlopeIntercept Form. vertical line passing through ( 5, ) = 5 Cannot be written in the slopeintercept form.. line passing through (, ) and ( ) 5, y = y = +. line passing through (, ) and ( 5, ) 5+ 8y = y = line passing through (, ) and (, 7) = Cannot be written in the slopeintercept form. 6. line parallel to + y = 6and passing through + y = 5 y = + 5 (, ) 7. line passing through ( 5, ) and parallel to y = 0 y = y = 0 8. line perpendicular to + 5y = and 5 y = 9 passing through (, 7 ) y = +
50 FUNCTIONS Recall that relation is a set of ordered pairs and that a function is a special type of relation. A function is a set of ordered pairs (a relation) in which to each first component, there corresponds eactly one second component. The set of first components is called the domain of the function and the set of second components is called the range of the function. Thus, the definition of function can be restated as: A function is a special type of relation. If each element of the domain corresponds to eactly one element of the range, then the relation is a function. Eamples: Determine if the following are functions. Domain Range y Domain Range y 7 5 This is a function, since for each first component, there is eactly one second component. This is not a function since for the first component, 7, there are two different second components, and 5. Eamples: Find the domain of the following functions.. 5 y = The denominator of a rational epression 0. Since + + 0,. The domain can be written as The domain is (, ) (, ). (, ) (, ).. y = Since this is a linear equation, there is an infinite number of values for, without restriction. The domain is any real number ( is any real number). The domain is (, ).. y = + The radicand of a square root must be 0. Since + 0,. The domain can be written as The domain is [, ). [, ). 6
51 FUNCTIONS Eamples: Find the range of the following functions.. y = Since is squared, the corresponding value for y must be 0. The range, 0 0,. The range is [ 0, ). y, can be written as [ ). y = Since this is a linear equation, there is an infinite number of values for y, without restriction. The range is any real number (y is any real number). The range is (, ).. y = + Since the square root yields an answer that is nonnegative, y 0. The range, 0, 0,. The range is [ 0, ) y can be written as [ ) Vertical Line Tests: A graph in the plane represents a function if no vertical line intersects the graph at more than one point. Eamples: Use the Vertical Line Test to determine if the graph represents a function.. y This is a graph of a function. It passes the vertical line test.. y (,) (,) This is not a graph of a function. For eample, the value of is assigned two different y values, and . Since we will often work with sets of ordered pairs of the form (,y), it is helpful to define a function using the variables and y. Given a relation in and y, if to each value of in the domain there corresponds eactly one value of y in the range, then y is said to be a function of. 7
52 FUNCTIONS Notation: To denote that y is a function of, we write y = f ( ). The epression f ( ) is read f of. It does not mean f times. Since y and f ( ) are equal, they can be y =, or we can write f ( ) used interchangeably. This means we can write =. Evaluate: To evaluate or calculate a function, replace the in the function rule by the given value from the domain and then compute according to the rule. Eample:. Given: f ( ) = 6+ 5 Find: f (, ) f ( 0, ) f ( ) f f f ( ) ( ) = = 7 ( ) ( ) 0= 60+ 5= 5 ( ) ( ) = 6 + 5= g = Given: ( ) Find: g(, ) g( 0, ) g( ) ( ) ( ) ( ) g = 5+ 8= 6 g g ( ) ( ) ( ) 0 = = 8 ( ) ( ) ( ) = = 6 8
53 FUNCTIONS Problems Answers. Determine whether or not each relation defines a function. If no, eplain why not. a. Domain Range y 8 0 a. Yes, for each first component, there corresponds eactly one second component b. Domain Range y { } c. (, ), (, ), (,0) b. No, for each first component,, there corresponds two different second components, and 7. c. No, for the first component,, there corresponds two different second components, and 0. { } d. (, ),(, ),(,) d. Yes, for each first component there corresponds eactly one second component.. Determine the domain and range of each of the following: { } a. (, ),(, ),( 0,0 ),(,7) a. Domain: {,,0,} Range: { 0,,,7 } b. y b c. y c Domain:{ 5, 7, 9} Range: {,,6 } Domain: {,5,9 } Range: { 7,, } 9
54 FUNCTIONS Problems Answers. Given ( ) ( ), f = + and g = find the following: a. f ( ) a. 0 b. f ( ) b. c. g ( 0) c. d. g ( ) d. 5. Given k( ) = and h( ) 5 =, find the following: a. k ( ) a. 9 b. k ( 5) b. 5 c. h ( ) c. d. h( 7) d
55 ALGEBRA AND COMPOSITION OF FUNCTIONS OPERATIONS ON FUNCTIONS If the domains and ranges of functions f and g are subsets of the real numbers, then: The sum of f and g, denoted as f + g, is defined by ( f + g)( ) = f ( ) + g( ) The difference of f and g, denoted as f g, is defined by ( f g)( ) = f ( ) g( ) The product of f and g, denoted as f g, is defined by ( f g)( ) = f ( ) g( ) The quotient of f and g, denoted as f / g, is defined by f ( ) ( f / g)( ) = where g( ) 0 g( ) The domain of each of the above functions is the set of real numbers that are in the domain of both f and g. In the case of the quotient, there is the further restriction that ( ) 0 g. The composite function f g is defined by ( f g)( ) = f ( g( ) ) 5
56 ALGEBRA AND COMPOSITION OF FUNCTIONS Problems Answers Let f ( ) = 6 5 and g( ) = + Find: ( f + g)( ) = ( f g)( ) = ( f g)( ) = f g ( ) = f ( g( )) = ( f g)( ) = g( f ( )) = ( g f )( ) =
57 EXPONENTS LAWS OF EXPONENTS Definition: n a = a a a... a n times a = the base n = the eponent Properties: Product Rule: m n m n a a = a + Eample: + a a = a a a a a = a = a 5 times times Quotient Rule: a a m n m n = a Eample: 8 = = 8 6 n Power Rule: ( ) m a nm = Eample: ( ) a y = y = y Raising a product to a power: ( ) n n n ab = a b Raising a quotient to a power: n a a = b b n n Zero Power Rule: 0 a = Eample: ( ) 0 ( ) = ; = = = Eamples: Simplify ab a b c = a b c = a b c. ( )( ) 8 6. = = y y 8y. y ( y+ 5y ) = ( y )( y) + ( y )( 5y ) = 6 y + 5y = 6y + 5y 5
58 EXPONENTS NEGATIVE EXPONENTS Definitions: a m m a = a m a = a m a a b m n m b a m n m Note: The act of moving a factor from the numerator to the denominator (or from the denominator to the numerator) changes a negative eponent to a positive eponent. Eamples: Simplify... 0 ab 0 = = 0 0, 000 ab a b a a = = = 5 y yz = = 5 z b b Since b has a negative eponent, move it to the denominator to become b Only the factors with negative eponents are moved.. y y = y y y = 6 ( ) 6 ( ) y = Within the parenthesis, move the factors with negative eponents. Simplify within the parenthesis. Use the power rule for eponents. ( ) y = = Move all factors with negative eponents. y ( ) 6y = Simplify. 5
59 EXPONENTS RATIONAL EXPONENTS Eamples: Simplify. 5 5 = 5 = 5 +. = = = = 6 = 6 = 6. ( ) = = 5. ( ) ( ) ( ) ( ) a b = a b = ab y 5 = y 5 5 y 5 = y = y 5 55
60 EXPONENTS Problems Answers Simplify. a a a ( a ) a 5. ( ) 6. ( ) a a 7. ( 6y ) 8y 8. ( ab ) ab 9. y y / y 56
61 RADICALS Definition of Square Root of a The square root of the number a is b, a b b a =, if =. The radical symbol,, represents the principal or positive square root. The number or variable epression under the radical symbol is called the radicand. Eamples: Simplify 5 = 5 Check: ( 5) = 5 = Check: ( ) = = Check: ( ) = 6 6 Properties: is not a real number If " " a is a negative real number, a is not a real number.. Product Rule: a b = ab and ab = a b Eamples: Simplify = 6 8 = =. Quotient Rule: a = a and a = a b b b b Eamples: Simplify 9 9 = = 8 8 = = = = 57
62 RADICALS Problems Answers Use the product rule and the quotient rule to simplify each radical. All variables represent positive real numbers y y y. 8 y y m 0 m y 5 y 58
63 RADICALS Definition: n th root of a. The n th root of the number a is b, n a = b if For n a, n is the inde number and a is the radicand. Eamples: Simplify = Check: ( ) 8 = 8 n b = a. 6 = Check: ( ) = 6 The product and quotient rules for radicals apply for all inde numbers and larger (for all n ). a b = ab and ab = a b n n n n n n n n n a a a a = and = n n n b b b b Eamples: Simplify. 8 = 6 = Notice that 6 is a perfect cube. That is, 6 =.. 80 = 6 5 = 5 Notice that 6 is a perfect th. That is, 6 =.. ( + y) = ( + y) ( + y) 5 ( y) ( y) = y z = y y z = y z y Notice that 0 5 0, y, and z are all perfect 5 th s That is, ( ), y ( y), and z ( z ) = = =. = yz y 5 59
64 RADICALS Problems Answers Simplify. All variables represent positive real numbers y y y 7. ( + y) 0 ( + y) 7 ( + y) a b c a b c ac b b 7. 6 y 7 y 60
65 RADICALS ADDITION/SUBTRACTION OF RADICALS To have like radicals, the following must be true:. The radicals must have the same inde.. The radicands must be the same. To combine like radicals (i.e. add or subtract);. Perform any simplification within the terms.. Use the distributive property to combine terms that have like radicals. Eamples: Perform the operations. Simplify.. + = ( + ) 6 6 = 8. = ( ) = 5 8 These radicals are like radicals. Add or subtract the coefficients, using the distributive property.. + = ( + ) = = 8 9 = 8 = = = = = = = = ( ) = Create like radicals first. Then combine like radicals = 7 = ( ) = 6
66 RADICALS Problems Answers Perform the operations. Simplify y 00y y 9. 5y y 6y 0. 00y 8y y 6
67 RADICALS Problems Answers 5. 0a a 0 a. 8 6 y y r s t 0 8 r s t 6 rt. 8a a 6a y z y z y b b b a b c ab c a c ( 5)( 5) + 9 a. ( a)( a). ( 5+ )( 5 ) 6
68 RATIONAL EXPONENTS Definition of a Rational Eponent m n a = n a m n ( a ) m Eamples: Simplify 6 = 6 = ( ) 8 = 8 = ( ) ( ) 6 = 6 = 8 = 5 ( ) ( ) 7 = 7 = = 9 5 = = = ( ) ( ) ( ) ( ) abc = a b c = ab c 6 ( 8y) = = = y 6 6 ( 8y) 8 ( ) ( y ) ab = a b Use Power Rule of Eponents: ( ) n n n 6
69 RATIONAL EXPONENTS Problems Simplify Answers ( ) 6 8 ab ab 9. ( ) 0 b 8 5 b 0. ( ) 8 8a b a b. ( ) p q p q 7 65
70 RADICAL EQUATIONS To solve a Radical Equation: S. Isolate one radical epression on one side of the equation.. Raise both sides of the equation to the power that is the same as the inde of the radical.. Solve the resulting equation. If it still contains a radical, go back to step.. Check the results to eliminate etraneous solutions. Eample: Solve = 9 = 9 ( ) = ( 9) = = 0+ 8 ( )( ) 0 = 6 = = 6 Solve. Isolate the radical. The inde number is, so square both sides. Check = Check = 6 Check for etraneous solutions. = 9 ( ) Answer: = 9 6 ( ) 6 = = 9 9 = is a solution and = 6 is an etraneous solution. 66
71 RADICAL EQUATIONS Eample: Solve + 7 = + 7 = + Isolate one radical ( 7) ( ) + = + The inde number is, so square both sides. + 7= + + 6= = ( ) = The inde number is, so square both sides. 9 = Isolate the radical. Check = = = Answer: = 9 67
72 RADICAL EQUATIONS Problems Answers Solve.. 5 = =, =. + = =, =. 9 = = 0 ( 7 is etraneous) = 5 No solution is etraneous 5. = = = = 7 68
73 COMPLEX NUMBERS The imaginary number i is defined as i =. From the definition, it follows that i =. Eample: Simplify 9 9= 9 = 9 = i = i Eample: Simplify = 6 = 6 = i 6 = i 6 A comple number is any number that can be written in the form a + bi, where a and b are real numbers and i =. Eample: Write this comple number in a + bi form: = = 5+ 9i Eample: Simplify and write in a + bi form: 9 9 = 9 = 7i = 0+ 7i 69
74 COMPLEX NUMBERS Problems Write each imaginary number in terms of i.. 5 Answers 5i. i or i. 08 6i or 6 i. 6 8 i Write each comple number in the form a + bi i i or 5+ i i i i i 0 or i i 6 or 6 6 i 70
75 COMPLEX NUMBERS OPERATIONS WITH COMPLEX NUMBERS Eample: Perform this addition. Write the answer in a + bi form. ( + 5) + ( 5 ) ( + 5) + ( 5 ) = ( + 5i) + ( 5 i) ( 5) ( 5i i) = + + = 8+ i = i Combine like term. Eample: Perform this multiplication. Write the answer in a + bi form. ( 6 )( + ) ( 6 )( + ) = ( i)( + i) = + 6i 6i 8i = 0i 8i ( ) = 0i 8 = 0 0i = i Distribute Property Combine like terms. Recall i = Eample: Perform this division. Write the answer in a + bi form. 5 7i 5 5 i = 7i 7i i 5i = 7i 5i = 7 ( ) 5 = 0 i 7 Recall i = 7
76 COMPLEX NUMBERS COMPLEX CONJUGATES The comple numbers a + bi and a bi are called comple conjugates. Eample: Perform this division. Write the answer in a + bi form. 6+ i 5i 6+ i 6+ i + 5i = 5i 5i + 5i = ( 6+ i)( + 5i) ( 5i)( + 5i) Multiply the numerator and the denominator by the conjugate of the denominator i+ i+ 0i = 9 + 5i 5i 5i Distributive Property 8 + i+ 0i = 9 5i Combine like terms. ( ) ( ) 8 + i + 0 = 9 5 Recall i = + i = = + i a + bi form = + i 7 7 7
77 COMPLEX NUMBERS Problems Answers Perform the indicated operations. Write the answer in a + bi form.. ( 5i) ( 0 i) i. ( 7 6 ) + ( ) 6 + i. 5i( 8 6i) i. ( 7i)( 5 i) i 5. ( 7 )( + 6) + i i 6 0 i i i i + i i i i + i 0 5i 7
78 SOLVING QUADRATIC EQUATIONS SOLVING QUADRATIC EQUATIONS USING THE SQUARE ROOT PROPERTY Eample: Solve 8 = 0 = 8 0 = 8 Isolate the perfect square. = ± 8 Find the square root of both sides using the Square Root Property. =± 9 =± Remember to use ± sign when finding the square roots. = ± The solutions are = and =. Eample: Solve ( ) = 6 ( ) = 6 ( ) =± 6 The perfect square is already isolated. Find the square root of both sides using the Square Root Property. =± = = = 7 = =± needs to be solved with two separate equations. The solutions are = 7 and =. 7
79 SOLVING QUADRATIC EQUATIONS Eample: Solve + = = 6 0 = 6 Isolate the perfect square. =± 6 Remember to use the ± sign when = ± 8i The solutions are = 8 i and = 8i. finding the square roots. Recall that = i. Eample: Solve = = 0 6 = 9 9 = 6 9 = ± 6 = ± = ± 7i = ± 9 6 ( ) 7 The solutions are 7 7 = i and = i. 75
80 SOLVING QUADRATIC EQUATIONS SOLVING QUADRATIC EQUATIONS USING SQUARE ROOT PROPERTY Problems Answers Solve the equations. Be sure to get two solutions.. = 6 = ±. = 5 = ± 5. = 0 = ± 0. = = ± 5. = 9 = ± 6. = ± = 00 0 i 7. = = ± i 6 8. = 50 = ± 5i 76
81 SQUARE QUADRATIC EQUATIONS SOLVING QUADRATIC EQUATIONS USING SQUARE ROOT PROPERTY Problems Answers = = 7 and = 5 9. ( ) 6 + = = 5 and = 9 0. ( ) 9 = = 0 and = 0. ( ) 5 5. ( + ) = 8 = ± + = = ± i. ( ) = = ± i. ( ) 9 5. ( 8) = 60 = 8 ± 5 6. ( ) 0 + = = ± i 0 7. ( ) 8 = = ± i 77
82 SOLVING QUADRATIC EQUATIONS SOLVING QUADRATIC EQUATIONS USING FACTORING The ZeroFactor Property When the product of two real numbers is 0, at least one of them is 0. If a and b represent real numbers, and if ab = 0 then a = 0 or b = 0 Eample: Solve using factoring 7 = = Set equation equal to 0. ( )( ) + = 0 Factor completely. + = 0 or = 0 Use the ZeroFactor property to set each factor equal to 0. + = 0 = 0 = = = Solve the resulting linear equations. Check: = Check: = 7 0 = 7 = 0 ( ) 7( ) = 0 7 = = 0 + = = 0 + = 0 0= 0 Check the results in the original equation. The solutions are = and =. 78
83 SOLVING QUADRATIC EQUATIONS SOLVING QUADRATIC EQUATIONS WITH THE QUADRATIC FORMULA Eample: Solve 7 = = Set the equation equal to zero. a=, b= 7, and c = Compare this equation with the Standard Form a + b + c = 0 and identify the a, b and c values in this equation. b ± b ac = a ± = ( 7) ( 7) ( )( ) ( ) ( ) + 7 ± 9 8 = 8 7 ± ± 8 = = 8 8 7± 9 = 8 Substitute the values for a, b and c into the Quadratic Formula. b ± b ac = a = = = = = = Check: = Check: = 7 = 0 7 = 0 ( ) 7( ) = 0 7 = = = = = 0 0= 0 Check the results in the original equation. The solutions are = and =. 79
84 SOLVING QUADRATIC EQUATIONS The Discriminant: For a quadratic equation of the form a + b + c = 0 with realnumber coefficients and a 0, the epression b ac is called the discriminant and can be used to determine the number and type of the solutions of the equation. Discriminant: b ac Number and type of solutions Positive 0... Negative.. Two different real numbers One repeated solution, a rational number Two different imaginary numbers that are comple conjugates 80
85 SOLVING QUADRATIC EQUATIONS Problems Answers Solve = 0 = 7 and =. = 0 = and =. = = and = = 0 = ± 6 5. = 5 = 0 and = = 6 = and = 7. 0 = = 6 and = = 0 = ± 8
86 SOLVING QUADRATIC EQUATIONS Problems Answers Solve = = ± = 0 = ± i. = = ± i. = 5 ± = = ± = 0 = 5 ±. = ± = = ± = 0 ± = = ± 8
87 SOLVING QUADRATIC AND RATIONAL INEQUALITIES TO SOLVE A QUADRATIC OR RATIONAL INEQUALITIES:. Write the inequality in standard form with 0 on the right side of the inequality.. Solve the related equation. The solutions are called critical numbers.. Consider any restrictions on the domain. For eample, in rational inequalities since the denominator cannot equal 0, set the denominator equal to 0 to determine the restrictions.. Locate the numbers from Steps and on a number line to create intervals. Determine if these numbers are included in the solution. 5. Test a number from each interval back in the original inequality. If it makes the inequality true, the interval is included in the solution. Eample: Solve the inequality 0 < < 0 < 0 0 Write the inequality in standard form. 0 0 = ( )( ) + 5 = 0 = = 5 Solve the related equation. (There are no restrictions on the domain)., 5,5 ( 5, ) Locate the numbers on a number line. Create the intervals. and 5 are not in the solution set. Eample continues on the net page. 8
88 SOLVING QUADRATIC AND RATIONAL INEQUALITIES Test a number from each interval into 0 <. test = test = 0 test = 6,,5 5 ( 5, ) Test a number from each interval in the original inequality. For interval,, test =. < 0 ( ) ( ) < 0 + < 0 6 < 0 False, this interval is not included in the solution set. For interval,5, test = 0. < 0 ( ) ( ) 0 0 < 0 0 < 0 True, this interval is included in the solution set. For interval ( 5, ), test = 6. < 0 ( ) ( ) 6 6 < 0 0 < 0 False, this interval is not included in the solution set. The solution is,5. 8
89 SOLVING QUADRATIC AND RATIONAL INEQUALITIES Eample: Solve the inequality = = = 0 ( ) ( ) = The inequality is in standard form. Solve the related equation. Consider restrictions on the domain. 5 Locate the numbers on a number line. (, ] [, 5) ( 5, ) Create the intervals. is in the solution set but 5 is not. For interval (, ], test = Test a number from each interval in the original inequality. True, this interval is included in the solution set. Eample continues on the net page. 85
90 SOLVING QUADRATIC AND RATIONAL INEQUALITIES For interval [, 5), test = False, this interval is not included in the solution set. For interval ( 5, ), test = True, this interval is included in the solution set. The solution is (, ] ( 5, ). 86
91 SOLVING QUADRATIC AND RATIONAL INEQUALITIES Solve the inequalities. Problem Answer. 6 6,. > 8 (, ) (, ). 5 < 0 + (, 5). + 0 [, ] (, ) 87
92 EQUATIONS THAT CAN BE WRITTEN IN QUADRATIC FORM SOLVING EQUATIONS USING USUBSTITUTION EXAMPLE: Solve for " " ( ) ( ) = 0 ( ) ( ) = 0 This equation is quadratic in form. u 5u+ 6= 0 Substitute " u " in the equation for + Let + = u. ( ) ( u )( u ) = 0 u = 0 u = 0 u = u = Factor and solve. + = + = = 0 = To solve the original equation for " ", substitute " + " in for " u " (Let u = + ). Check = 0 ( ) ( ) = 0 ( ) ( ) 5 ( ) = 0 Check = ( ) ( ) = 0 ( ) ( ) ( ) = 0 The solutions are = 0 and =. 88
93 EQUATIONS THAT CAN BE WRITTEN IN QUADRATIC FORM SOLVING EQUATIONS USING USUBSTITUTION Eample: Solve for " " + 7 8= = 0 This equation is quadratic in form = 0 ( ) u + 7u 8= 0 Substitute " u " in the equation for Let = u ( ) ( u )( u ) + 8 = 0 u = 0 u+ 8= 0 u = u = 8 Factor and solve = = 8 = = 8 = = To solve the original equation for " ", substitute in for " u ". Let u = ( ) Check = Check = = = 0 ( ) ( ) = 0 ( ) ( ) ( ) = 0 The solutions are = and =. 89
94 EQUATIONS THAT CAN BE WRITTEN IN QUADRATIC FORM Problems Answers Solve using USubstitution.. ( ) ( ) = 0 = and = 0. ( ) ( ) = 0 = and = 5. + = 0 =± and =± i = 0 = and = = 0 = and = = 0 = 7,776 and = 90
95 GRAPHING A QUADRATIC FUNCTION OF THE FORM ( ) = f a b c ( a ) The graph of f ( ) = a + b + c is a parabola. Use the coefficients a, b, and c to find the following:. The parabola opens upward when a > 0. The parabola opens downward when a < 0.. The coordinate of the verte of the parabola is verte is b b b f. The verte is, f a a a. b =. The ycoordinate of the a. The ais of symmetry is the vertical line passing through the verte.. The yintercept is determined by the value of ( ) ( 0, f ( 0) ) or ( 0, c ). f when = 0 : the yintercept is 5. The intercepts (if any) are determined by the values of that make f ( ) = 0. The  intercept/s, if any, is (,0). To find them, solve the quadratic equation a + b + c = 0. 9
96 GRAPHING A QUADRATIC FUNCTION OF THE FORM f = a + b + c ( a ) f = 8+ 6 Eample: Graph ( ) ( ) 0 a=, b= 8, c= 6 Determine the values of a, b, and c.. a = ; parabola opens upwards The parabola opens upward ( 0) a >.. 8 = = f f ( ) ( ) ( ) ( ) ( ) = = 8+ 6 b a The verte is, f = (, ) b The coordinate of the verte is =. a b The ycoordinate of the verte is f. a. = is the ais of symmetry. Ais of symmetry is the vertical line passing through the verte.. f ( 0) = 0 ( ) 80 ( ) + 6 f ( 0) = 6 The yintercept is ( 06, ) 5. ( ) f = = 8+ 6 ( ) 0= + ( )( ) 0= = 0 = 0 = = Let = 0. The yintercept is ( 0, ( 0) ) f. Let f ( ) = 0. The intercept/s, if any, is (,0). The intercepts are ( 0, ) and (, ) y Notes ais of symmetry: = yintercept: (0,6) intercepts: (,0) and (,0) 9
97 GRAPHING A QUADRATIC FUNCTION OF THE FORM f = a + b + c ( a ) ( ) 0 Problems Answers Use the 5step process for graphing these quadratic functions.. ( ) f = + 6 y Step.Opens upward since > 0. Step. Verte is (, 8). Step. Ais of symmetry is =. 0, 6. Step. yintercept is ( ) Step 5. The intercepts are (,0) and ( ), 0.. ( ) f = 5 Step. Opens upward since > 0. y Step. Verte is (, 6). Step.Ais of symmetry is =. 0, 5. Step. yintercept is ( ) Step 5. intercepts are ( 7,0 ) and ( 5,0).. ( ) f = + Step. Opens downward since < 0. y Step. Verte is (,). Step. Ais of symmetry is =. Step. yintercept is ( 0, ). Step 5. intercepts are (,0) and (,0). 9
98 PYTHAGOREAN THEOREM In a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the legs (the sides that form the right angle). If c is the hypotenuse and a and b are the lengths of the legs, this property can be stated as: c b c = a + b or c= a + b a a = c b or a= c b b = c a or b= c a Eamples: Find the length of the unknown side in the following right triangles.. Given a= and b= 5, find c. c = c =. Given b= and c=, find a. a = 69 a = 5 a = 5. Given c= 6 and a= 5, find b. b = 6 5 b = 9
Polynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationSUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills
SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationBig Bend Community College. Beginning Algebra MPC 095. Lab Notebook
Big Bend Community College Beginning Algebra MPC 095 Lab Notebook Beginning Algebra Lab Notebook by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationModuMath Algebra Lessons
ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationMathematics Placement
Mathematics Placement The ACT COMPASS math test is a selfadaptive test, which potentially tests students within four different levels of math including prealgebra, algebra, college algebra, and trigonometry.
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More information2. Simplify. College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses
College Algebra Student SelfAssessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationMethods to Solve Quadratic Equations
Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a seconddegree
More informationMyMathLab ecourse for Developmental Mathematics
MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and
More informationAlgebra 12. A. Identify and translate variables and expressions.
St. Mary's College High School Algebra 12 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used
More informationSkill Builders. (Extra Practice) Volume I
Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining
More informationALGEBRA I / ALGEBRA I SUPPORT
Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.
More informationCRLS Mathematics Department Algebra I Curriculum Map/Pacing Guide
Curriculum Map/Pacing Guide page 1 of 14 Quarter I start (CP & HN) 170 96 Unit 1: Number Sense and Operations 24 11 Totals Always Include 2 blocks for Review & Test Operating with Real Numbers: How are
More informationSECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS
(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. 1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationAlgebra 1 Course Objectives
Course Objectives The Duke TIP course corresponds to a high school course and is designed for gifted students in grades seven through nine who want to build their algebra skills before taking algebra in
More informationCOGNITIVE TUTOR ALGEBRA
COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One intercept and all nonnegative yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More informationStudents will be able to simplify and evaluate numerical and variable expressions using appropriate properties and order of operations.
Outcome 1: (Introduction to Algebra) Skills/Content 1. Simplify numerical expressions: a). Use order of operations b). Use exponents Students will be able to simplify and evaluate numerical and variable
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationModule: Graphing Linear Equations_(10.1 10.5)
Module: Graphing Linear Equations_(10.1 10.5) Graph Linear Equations; Find the equation of a line. Plot ordered pairs on How is the Graph paper Definition of: The ability to the Rectangular Rectangular
More informationSECTION P.5 Factoring Polynomials
BLITMCPB.QXP.0599_4874 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More information( ) ( ) Math 0310 Final Exam Review. # Problem Section Answer. 1. Factor completely: 2. 2. Factor completely: 3. Factor completely:
Math 00 Final Eam Review # Problem Section Answer. Factor completely: 6y+. ( y+ ). Factor completely: y+ + y+ ( ) ( ). ( + )( y+ ). Factor completely: a b 6ay + by. ( a b)( y). Factor completely: 6. (
More informationMultiplying Polynomials 5
Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationA.3. Polynomials and Factoring. Polynomials. What you should learn. Definition of a Polynomial in x. Why you should learn it
Appendi A.3 Polynomials and Factoring A23 A.3 Polynomials and Factoring What you should learn Write polynomials in standard form. Add,subtract,and multiply polynomials. Use special products to multiply
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationNorwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction
1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K7. Students must demonstrate
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationFactoring and Applications
Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationSect 6.7  Solving Equations Using the Zero Product Rule
Sect 6.7  Solving Equations Using the Zero Product Rule 116 Concept #1: Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0 (referred
More informationRational Expressions and Rational Equations
Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple
More informationFlorida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies  Lower and Upper
Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies  Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,
More information7.1 Graphs of Quadratic Functions in Vertex Form
7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called
More informationHIBBING COMMUNITY COLLEGE COURSE OUTLINE
HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE:  Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
More information12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2
DMA 080 WORKSHEET # (8.8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) 00 ) 8 27 2/ Use a calculator to approximate the square root to decimal
More informationMATH 100 PRACTICE FINAL EXAM
MATH 100 PRACTICE FINAL EXAM Lecture Version Name: ID Number: Instructor: Section: Do not open this booklet until told to do so! On the separate answer sheet, fill in your name and identification number
More informationDevelopmental Math Course Outcomes and Objectives
Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/PreAlgebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information1.1 Practice Worksheet
Math 1 MPS Instructor: Cheryl Jaeger Balm 1 1.1 Practice Worksheet 1. Write each English phrase as a mathematical expression. (a) Three less than twice a number (b) Four more than half of a number (c)
More informationVeterans Upward Bound Algebra I Concepts  Honors
Veterans Upward Bound Algebra I Concepts  Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER
More informationBrunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year.
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 20142015 school year. Goal The goal of the summer math program is to help students
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationDMA 060 Polynomials and Quadratic Applications
DMA 060 Polynomials and Quadratic Applications Brief Description This course provides a conceptual study of problems involving graphic and algebraic representations of quadratics. Topics include basic
More informationWarmUp Oct. 22. Daily Agenda:
Evaluate y = 2x 3x + 5 when x = 1, 0, and 2. Daily Agenda: Grade Assignment Go over Ch 3 Test; Retakes must be done by next Tuesday 5.1 notes / assignment Graphing Quadratic Functions 5.2 notes / assignment
More informationMidterm 1. Solutions
Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationThe program also provides supplemental modules on topics in geometry and probability and statistics.
Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students
More informationMATH 10034 Fundamental Mathematics IV
MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.
More informationMTH 100 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created June 6, 2011
MTH 00 College Algebra Essex County College Division of Mathematics Sample Review Questions Created June 6, 0 Math 00, Introductory College Mathematics, covers the mathematical content listed below. In
More information25 Rational Functions
5 Rational Functions Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any 1 f () = The function is undefined at the real zeros of the denominator b() = 4
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationThis unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide.
COLLEGE ALGEBRA UNIT 2 WRITING ASSIGNMENT This unit has primarily been about quadratics, and parabolas. Answer the following questions to aid yourselves in creating your own study guide. 1) What is the
More informationCOMPETENCY TEST SAMPLE TEST. A scientific, nongraphing calculator is required for this test. C = pd or. A = pr 2. A = 1 2 bh
BASIC MATHEMATICS COMPETENCY TEST SAMPLE TEST 2004 A scientific, nongraphing calculator is required for this test. The following formulas may be used on this test: Circumference of a circle: C = pd or
More informationName Date Block. Algebra 1 Laws of Exponents/Polynomials Test STUDY GUIDE
Name Date Block Know how to Algebra 1 Laws of Eponents/Polynomials Test STUDY GUIDE Evaluate epressions with eponents using the laws of eponents: o a m a n = a m+n : Add eponents when multiplying powers
More informationAlgebra 2 YearataGlance Leander ISD 200708. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks
Algebra 2 YearataGlance Leander ISD 200708 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks
More informationMATD 0390  Intermediate Algebra Review for Pretest
MATD 090  Intermediate Algebra Review for Pretest. Evaluate: a)  b)  c) () d) 0. Evaluate: [  (  )]. Evaluate:  (7) + (8). Evaluate:   + [6  (  9)]. Simplify: [x  (x  )] 6. Solve: (x +
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationAlgebra 1. Curriculum Map
Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More information6.1 Add & Subtract Polynomial Expression & Functions
6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
More information