Advanced Algebra Name Date: Semester Final Exam Review # ) Given f x x, determine the average rate of change between and 6? fb fa f x 6 6 f 9 Completed Example: ba f 4 7 6, 9, 7 9 7 6 f x 6 8 Unit : Linear Functions & Regression f x a) Given, x determine the average rate of change between and 6? You Try: b) Given f x 4x, determine the average rate of change between and? ) Graph: y x a) Graph: y x b) Graph: y x 5 4 Steps: a) Plot y-int (0,-) b) move up, right What does the equation tell us? Point (other than ) What does the equation tell us? Point (other than y int) What does the equation tell us? Point (other than y int) ) Graph: y x Steps: a) Plot point (-, ) b) move up, right. a) Graph: y 4 x b) Graph: y 6 x What does the equation tell us? Point (other than y int) 5 What does the equation tell us? Point (other than ) 4) Graph: 4x y Steps: What does the equation tell us? Point (other than y int) 4a) Graph: x 9y 8 4b) Graph: 0x 5y 5 a) Find the x-intercept 4x 0 x, 0 b) Find the y-intercept 4(0) y y 0, - What does the equation tell us? Point (other than ) What does the equation tell us? Point (other than ) What does the equation tell us? Point (other than )
5) Write the equation of the line through (,0) & (,4) : 4 0 4 6 Choose an ordered pair and use the point-slope equation : y y mx x Equation : y 0 x or y 4 (x ) y x or y 4 (x ) 6) Write the equation of the line through (, 5), perpendicular to y x 5 opposite reciprocal slope 5a) Write the equation of the line through (0,5) & (,) Equation:. 6a) Write the equation of the line through ( 4,), parallel to y x 4 5b) Write the equation of the line through (, 4) & (, ) Equation:. 6b) Write the equation of the line through ( 4,4), perpendicular to y 4x 7 Old : New : Use the ordered pair and use the point-slope equation y y m x x 7) Write an equation of the line in three different forms. Steps: a) Find the y-intercept 0, b y 5 x y 5 x Equation: 7a) Write an equation of the line. Equation: 7b) Write an equation of the line. b) Calculate the slope Up, Right m c) Fill in SLOPE INTERCEPT FORM y mx b y x d) Fill in POINT SLOPE FORM y y mx x m Point, 4 y 4 x e) Convert from another form to STANDARD FORM y x x y x y * Answers may vary! Ax By C - Intercept Form: Point - Form: Standard Form: - Intercept Form: Point - Form: Standard Form:
8) The table shows the sales of a company (in million dollars) from 995 to 999. Use the data to determine approximately how many millions of dollars the company will make from sales in the year 05? Let x 0 represent the year 995. Year 995 996 997 998 999 Sales (millions) 9 9 7 45 8a) The table shows the number of insured commercial banks (in thousands) in the United States from 987 to 99. Use the data to determine approximately how many thousands of commercial banks would be insured in 05? Let x 0 represent the year 985. Year 987 988 989 990 99 Insured.7..7..9 y = $79,600,000 in sales Scan for calculator directions!! 9) How many solutions does the following system have? x y 4 9x 6y x y 4 9x 6y 9x 6y 0 0 inf initely many solutions Unit : Linear Systems 9a) How many solutions does the following system have? 6x 4y 0 x 8y 9b) How many solutions does the following system have? y x 4 8x 4y 6 0) For the given system, what is 4x y the value of x? x 5y 5 4x y x 5y 0a) For the given system, what is the value of y? 7x 5y 5 x 0y 0 0b) For the given system, what is the value of x? 4x y 6 x 4y 7 0x 5y 65 x 5y 8x 54 x ) Solve the system: x y 4z 8 x y z 5x y 6z 7 * apps plysmlt (or plysmlt) * enter again * : simult eqn solver * number of equations x number of unknowns * Next (graph) * Enter coefficients (0 for missing terms) * Solve (graph) (,, ) a) Solve the system: 6x y z 4 x 5y 6z 9 x 5y 6z 5 b) Solve the system: a 6b 4c 6a b 5c a 4b 5c 8
) Your local Boy s Scout Troop sold a total of 5 boxes of popcorn. They sold boxes of Caramel popcorn for $ each and boxes of White Chocolate for $4 each. The troop raised a total of $0. How many of each box of Caramel corn did your troop sell? x boxes of caramel popcorn y boxes of white chocolate popcorn a) The admission fee for the Taste of Lombard is $.50 for children and $4.00 for adults. On the first day,,00 people attended the Taste and $5,050 was collected. How many children attended? b) A local Girl s Scout Troop sold a total of 5 boxes of cookies. They sold boxes of Thin Mint Cookies for $ each and boxes of Caramel Delights for $ each and raised a total of $44. How many box of Thin Mints did the troop sell? x 4y 0 x y 5 Use plysmlt on your calc to solve (#) 0 boxes of caramel corn ) Jocelyn and two friends are preparing for a party so they decide to stock up on candy. Carina spends a total of $7.88, Nick spends $.88 and Jocelyn spends $56.8. The table shows the amounts of jelly beans, chocolates and caramels that each person purchased. What is the price per pound for chocolates? Jelly Beans Chocolates Caramels Carina lb lb 0.5 lb Nick lb lb 0.5 lb Jocelyn 0.5 lb lb 5 lb x price per lb of jelly beans y price per lb of chocolates z price per lb of caramels x y 0.5z 7.88 x y 0.5z.88 0.5x y 5z 56.8 Use plysmlt on your calc to solve (see #).5, 6.50, 7.5 a) Uriel and two friends buy snacks for a field trip. Uriel spends a total of $8, Sal spends $9 and Jon Anthony spends $9. The table shows the amounts of mixed nuts, granola and dried fruit that each person purchased. What is the price per pound of granola? Mixed Nuts Granola Dried Fruit Uriel lb 0.5 lb lb Sal lb 0.5 lb 0.5 lb Jon Anthony lb lb 0.5 lb b) Three students invested their money to earn interest. Each student divided the same amount of money into three different types of accounts at different banks. At the end of the year, Marco had made $87.50 in interest, James made $59.00 and Alexa made $0.00. How much money did the students invest in the savings bonds? Savings Account Annual Interest CD Savings Bonds Marco 4% 5.5% 6% James.5% 6% 4.% Alexa 5% 4% 7% $6.50 per pound of chocolates 4) Write the system of inequalities that fits the following graph: 4a) Write the system of inequalities that fits the following graph: 4b) Write the system of inequalities that fits the following graph: y x y x
5) Given the following graph and objective quantity, find the location (ordered pair) of the maximum value of the objective quantity. Objective Quantity: P 4x 8y 5a) Given the following graph and objective quantity, find the location (ordered pair) of the maximum value of the objective quantity. Objective Quantity: P.5x 6y 5b) Given the following graph and objective quantity, find the location (ordered pair) of the maximum value of the objective quantity. Objective Quantity: P x y (0,5) (4,5) (0,) (,4) (0,0) (8,0) (0,0) (6,0) P 4(0) 8(5) 40 P 4(0) 8(0) 0 P 4(8) 8(0) P 4(4) 8(5) 56 6) Yum s Bakery bakes two breads, Italian and Whole Wheat. They want to maximize their profits from bread sales. One batch of Italian yields a profit of $40. One batch of Whole Wheat yields a profit of $0. One batch of Italian uses 5 pounds of oats and pounds of flour. One batch of Whole Wheat uses pounds of oats and pounds of flour. The company has 80 pounds of oats and 5 pounds of flour available. Write the equation for the objective quantity and determine the constraints. Objective Quantity : P 40a 0b Constraints : x 0 y 0 5x y 80 x y 5 (4,5) oats and flour are not negative constra int for lbs of OATS constra int for lbs of FLOUR 6a) A book store manager is purchasing new bookcases and trying to minimize cost. Bookcase A costs $00 and Bookcase B costs $5. The store needs 0 feet of shelf space. Bookcase A provides ft of shelf space and Bookcase B provides 6 ft of shelf space. Due to of space restrictions, the store has room for at most 8 of bookcase A and of bookcase B. Write the equation for the objective quantity and determine the constraints. Objective Quantity: Constraints: 6b) Vanessa makes bracelets and necklaces to sell at a craft store and she is trying to maximize her profits. Each bracelet makes a profit of $7, takes hour to assemble, and costs $ for materials. Each necklace makes a profit of $, takes hour to assemble, and costs $ for materials. Vanessa has 48 hours available to assemble bracelets and necklaces and she has $78 available to pay for materials. Write the equation for the objective quantity and determine the constraints. Objective Quantity: Constraints: 7) Solve for x: 9x Positive Case : 9 x x x, 7 Unit : Absolute Value Equations, Inequalities, & Functions Negative Case : 9 x x x 7 7a) Solve for x: 8x 8 7b) Solve for x: 9x 9 6
8) Solve & graph: 4x 0 0 8a) Solve : x 9 8b) Solve: 64x Less thand 0 4x 0 0 0 4x 0 0 x 5 5 x 0-5 0 5, 0 9) Given the equation y x, determine all transformations. Reflect over the x - axis, stretch, Right, Up 0) Given the equation y x, determine the vertex and whether it s a maximum or minimum on the graph. Vertex, a 0 Minimum ) Write the inequality for the graph. 9a) Given the equation y x 5 8, determine all transformations. 0a) Given the equation y x 5 8, determine the vertex and whether it s a maximum or minimum on the graph. a) Write the inequality for the graph. 9b) Given the equation y x 4 6, determine all transformations. 0b) Given the equation y x 4 6, determine the vertex and whether it s a maximum or minimum on the graph. b) Write the inequality for the graph. (,4) Vertex form: y ax ( h) k y ( x ) 4 ) Solve: x x 0 0 x x 0 0 4 5 (x 4)( x 5) 0, 5 4 Unit 4: Quadratic Functions a) Solve: x x 6 0 b) Solve: 5x 9x 6 ) Solve: 4x 8x 0 4 xx ( ) 0 0, a) Solve: x 0x 0 b) Solve: x 4 x 4) Solve the equation: ( x ) 4a) Solve the equation: 4( x ) 00 4b) Solve the equation: 9( x ) 8 ( x ) 4 x x 0, 4
5) Solve: 9x x 9x x 0 not factorable - use quad. formula 5a) Solve: 4x 4 x 5b) Solve: x 6x ( ) 4(9)( ) 80 x (9) 8 6 5 6 5 5 8 8 6) Simplify: 48 6 4i 7) State the number and type of solutions for the equation. x 4x 8x 6 x 6x 9 0 discri min ant : b 4ac 6a) Simplify: 7 6b) Simplify: 50 7.) State the number and type of solutions for the equation. x 5x 6 x 0 7b) State the number and type of solutions for the equation. x 4 x (6) 4()(9) 0 real rational solution 8) The graph of a quadratic equation is shown below. Use the graph to identify the solutions of the equation. x int ercepts : (, 0) and (4, 0) 8a) The graph of a quadratic equation is shown below. Use the graph to identify the solutions of the equation. 8b) The graph of a quadratic equation is shown below. Use the graph to identify the solutions of the equation. Solutions :, 4 9) Describe the leading coefficient in the equation fx ( ) ax bx c if the graph opens up. a 0 0) What are the coordinates of the minimum of this function? fx ( ) x x 5 Vertex : b, f b a a x f () 4 () () 5 (, ) 9a) Describe the leading coefficient in the equation fx ( ) ax bx c if the graph opens down. 0a) What are the coordinates of the minimum of this function? y x 6x 9b) Describe the leading coefficient in the equation fx ( ) ax bx c if the graph stretches. 0b) What are the coordinates of the maximum of this function? y x 8x 5
) Graph the function fx ( ) x 4x Use the formula above or complete the square x 4 4 () f () 4() Vertex: (, ) Parabola Pattern: Out, up Out, up 4 ) What are the s of the graph of this function? y x x 0 ( x x 6) 0 ( x )( x ) x, x intercepts : (, 0), (, 0) ) Write a possible equation (in vertex form) for the given graph: a) Graph the function fx ( ) x 8x 6 a) What are the s of the graph of this function? y x 4x 64 a) Write a possible equation (in vertex form) for the given graph: b) Graph the function fx ( ) x 4x b) What are the s of the graph of this function? y x 5x 8 b) Write a possible equation (in vertex form) for the given graph: Vertex : ( h, k) (,4) Vertex form : y ( x h) k y ( x ) 4 4) Rewrite the following into vertex form and find the vertex: y 9x 8x 55 y x x 9 55 b * 9 Vertex form: y 9 x 64 vertex :, 64 9 5) Use the graph to identify key characteristics of the function: f x x 6x 7 Domain : Range :,, Increasing :, Decreasing :, 4a) Rewrite the following into vertex form and find the vertex: y x x 4 64 5a) Use the graph to identify key characteristics of the function: Domain: Range: Increasing: Decreasing: f x x 8x 9 4b) Rewrite the following into vertex form and find the vertex: y x 4x 0 5b) Use the graph to identify key characteristics of the function: Domain: Range: Increasing: Decreasing: f x x 4x
6) Given the function f x x 6x 7, determine the average rate of change for the interval,. f x fa fb ba f 6 7 6 7 (, ) f 6 7 98 7 (, ) y y 4 x x 7) Determine the solution(s) to the following system of equations. 6a) Given the function above, f x x 8x 9, determine the average rate of change for the interval 4,. 7a) Determine the solution(s) to the following system of equations. 6b) Given the function above, f x x 4x, determine the average rate of change for the interval,. 7b) Determine the solution(s) to the following system of equations. Solution(s): ( 0, - ) & (, ) Solution(s): Solution(s): 8) Graph the system of inequalities. Shade the appropriate solution set. x x f x 4 g x 8a) Graph the system of inequalities. Shade the appropriate solution set. x x f x g x 5 8b) Graph the system of inequalities. Shade the appropriate solution set. x x f x g x 7 Solid Shade above Dashed Shade below
9) Jason jumped off of a cliff into the ocean in Acapulco while vacationing with some friends. His height as a function of time could be modeled by the function f x 6x 6x 480 where x is the time in seconds and f is the height in feet. a) What is the maximum height Jason reaches? Maximum = Vertex x y 6 6 ( 6) 6 6 480 484 ft 9a) A dud missile is fired straight into the air from a military instillation. The missile s height is given by the formula f x 6x 400x 00 where x is the time in seconds and f is the height in feet. a) What is the maximum height of the dud missile? 9b) A rock is thrown upward from the top of a tower. The height of a rock above the ground as a function of time can be modeled by the f x 6x 00x 5 where equation x is the time in seconds and f is the height in feet. a) What is the maximum height of the rock? b) How high above the ocean is Jason when he jumps? Initial hight is y - intercept: 480 ft b) How high above the ground is the military instillation? b) How high above the ground is the rock when it is thrown from the tower? 40) Describe the end behavior of the graph of 5 fx ( ) x 4x 5x 4. Positive Odd f x as x, as x, f x 4) Determine if the graph is a function: Unit 5: Polynomial Functions 40a) Describe the end behavior of the graph of 4 fx ( ) 4x 4x x. 4a) Determine if the graph is a function: 40b) Describe the end behavior of the graph of 9 4 fx ( ) x 6x x 5. 4b) Determine if the graph is a function: Yes, it passes the vertical line test 4) Factor: 5x 8x 5x 0 5x 8x 5x 0 x 5x 6 5 5x 6 5x 6x 5 4a) Factor: 4x 4x 5x 0 4b) Factor: 8m 5 xm 4 4mn 7xn
4) Use the graph to identify key characteristics of the function: 4a) Use the graph to identify key characteristics of the function: 4b) Use the graph to identify key characteristics of the function: 4 8 6 5 f x x x 4x 6 f x x x x f x x x x 9 4 9 s Increasing: Decreasing: 4,. Local Min: y 448 Local Max: Positive Interval: Negative Interval:, 4.,. y 5 5, 6. 08.,, 5 6., 08. Increasing: Decreasing: Local Min: f x 0 : f x 0 : Local Maxima: Increasing: Decreasing: Local Min: f x 0 : f x 0 : Local Max: 44) Identify the following: 44a) Identify the following: 44b) Identify the following: Degree: Even or Odd One end rises, the other falls Leading Coefficient: Positive or Negative Right end rises Real Zeros: 4 5 x- intercepts Possible Degree: 4 5 4 turns 45) Divide: (x 9x 6) (x 9) 7x 8 x 9 x 9x 6 x 6x 4x 6 4x 7 0 7x 8 0 x 9 Degree: Even or Odd Leading Coefficient: Positive or Negative Real Zeros: 4 5 Possible Degree: 4 5 45a) Divide: (8y y ) (4y ) Degree: Even or Odd Leading Coefficient: Positive or Negative Real Zeros: 4 5 Possible Degree: 4 5 45b) Divide: 4 (r 9r r 6r ) (r ) 46) Divide : ( n 6n n 8) ( n ) 6 8-5 8 5 8 0 n 0 n 46a) Divide : ( x x 6x 69) ( x 6) 46b) Divide: ( x x 59x 5) ( x 9) n 5n 8
47) If is a zero of f( x) x x 9x 9, find the other zeros. 9 9 9 4 0 47a) If is a zero of f( x) x 4x 4x 6, find the other zeros. 47b) If is a zero of f ( x) x 7x 7x, find the other zeros. ( x )( x 4x ) 0 ( x )( x )( x ) 0 x,,, 48) Write the polynomial function of least degree that has zeros of 5, i and i. ( x 5)( x i)( x i) f x x x x ( 5)( 4 ) x x xi xi i f ( 5)( 4 ) x x i f ( x 5) x 4 f x 4x 5x 0 f x 48a) Write the polynomial function of least degree that has zeros of 4, i and i. 48b) Write the polynomial function of least degree that has zeros of, i and i. fx x x x ( ) 5 4 0 Date Provided Due Date Homework - Monday 4 Column A Final Review # Tuesday 5 Column B Final Review # ** Extra Credit Due Monday Also** Period : Your Test is Wednesday /6 Period 8: Your Test is Friday /8