Advanced Algebra Name Date: Semester 1 Final Review ( ) , determine the average rate of change between 3 and 6? 4a) Graph: 3x

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1 Advanced Algebra Name Date: Semester Final Review ) Given f ( x) x = +, determine the average rate of change between and 6? f x = fa fb b a ( 6) = ( 6 ) + ( ) = ( ) f = + = 9 f + = 4 + = 7 ( 6, 9 ) (, 7 ) f x Completed Example: = = = 6 8 ( ) Unit : Linear Functions & Regression a) Given f ( x) = 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 ) ) Graph: y ( x ) Steps: a) Plot point (-, ) b) move up, right = +. What does the equation tell us? Point (other than y int) What does the equation tell us? Point (other than y int) 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 )

2 5) Write an equation of the line in three different forms. Steps: a) Find the y-intercept ( 0, ) b = 5a) Write an equation of the line. 5b) 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 = m( x + x ) Point, 4 m = ( x ) y 4 = e) Convert from another form to STANDARD FORM y = x + x + y = x y = Ax + By = C 6) Write the equation of the line through (,0) & (,4) : = = 6 Choose an ordered pairs to fill in y y = m( x x ) Equation : y 0 = x or y 4 = (x ) y = x or y 4 = (x + ) 7) Write the equation of the line through (, 5), perpendicular to y = x 5 Answers may vary opposite reciprocal - Intercept Form: Point - Form: Standard Form: 6a) Write the equation of the line through (0,5) & (,) Equation:. 7a) Write the equation of the line through ( 4,), parallel to y = x 4 - Intercept Form: Point - Form: Standard Form: 6b) Write the equation of the line through (, 4) & (, ) Equation:. 7b) Write the equation of the line through ( 4,4), perpendicular to y = 4x + 7 Old : New : Use the ordered pair to fill in: y 5 x = y + 5 = x + ( x x) y y = m Equation: Equation:

3 8) Using the points provided, perform a regression to find the equation of the best fitting line. Then find the indicated value. (4,54.5), (6, 66.4), (8, 8.9), (0,07), (, 8.7), (4,7.5). Find y when x = 0. * Stat * Edit (enter data) * y = (turn on plot and clear equations) * Window (set window) * Stat Calc, #4 LinReg * Vars y vars, enter until you get an equation y =.5x * graph * Trace and the down arrow * type in the x -value, enter 8a) Using the points provided, perform a regression to find the equation of the best fitting line. Then find the indicated value. (64,45), (68, 70), (74, 90), (6,0), (77, 0), (64, 8) Find y when x = 70. 8b)Using the points provided, perform a regression to find the equation of the best fitting line. Then find the indicated value. (0, 7), (, ), (, ), (, 4), (4, 44) Find y when x = 5. Answer : y = 8.8 ** Be sure your window will allow you to see x =0. ** 9) The table shows the sales of a company (in million dollars) from 995 to 999. If this trend continues, approximately how many millions will the company make from sales in the year 05? Year Sales (millions) a) The numbers of insured commercial banks (in thousands) in the United States for the years 987 to 99 are shown in the table. If this trend continues, approximately how many thousands of commercial banks would be insured in 05? Year Insured Follow steps from #8, remember to make sure your window will allow you to see x = 05 y = $79,600,000 in sales 0) How many solutions does the following system have? x + y = 4 9x + 6y = Unit : Linear Systems 0a) How many solutions does the following system have? 6x 4y = 0 x 8y = 0b) How many solutions does the following system have? y = x+ 4 8x 4y = 6 ( x + y = 4) 9x + 6y = 9x 6y = 0 = 0 inf inite solutions

4 ) For the given system, what is the 4x + y = value of x? x 5y = 5 ( 4x + y = ) x 5y = 0x + 5y = 65 x 5y = 8x = 54 x = ) Solve the system: x y 4z = 8 x + y + z = 5x y 6z = 7 a) For the given system, what is the value of y? 7x + 5y = 5 x 0y = 0 a) Solve the system: 6x + y + z = 4 x + 5y + 6z = 9 x 5y + 6z = 5 b) For the given system, what is the value of x? 4x + y = 6 x + 4y = 7 b) Solve the system: a + 6b 4c = 6a b + 5c = -a + 4b 5c = 8 On Calc MATRIX : *nd, matrix, edit * (equations) x 4 (terms) *Enter coefficients *nd, mode *nd, matrix, math, "rref" *nd, matrix, :, enter or POLYSMLT : * 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) (,, ) ) 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 popcorn 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 and how many adults 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 of each box of cookies did the troop sell? x + 4y = 0 x + y = 5 multiply by to elim. x x + 4y = 0 x y = 05 y = 5 x + 5 = 5 5 x 5 = 0 0 boxes of caramel corn 5 boxes of white chocolate popcorn

5 ) 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 of each type of candy? Jelly Beans 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 = x + y + 5z = 56.8 $7.5 per pound of caramels Chocolates Caramels On Calc MATRIX : POLYSMLT : *nd, matrix, edit * apps plysmlt (or plysmlt) * (equations) x 4 (terms) * enter again *Enter coefficients or * : simult eqn solver *nd, mode * number of equations *nd, matrix, math, "rref" x number of unknowns *nd, matrix, :, enter * Next (graph) * Enter coefficients (0 for $.5 per pound of jelly beans missing terms) * Solve (graph) $6.50 per pound of chocolates 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 each type of snack? 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 each type of account? Savings Account Annual Interest CD Savings Bonds Marco 4% 5.5% 6% James.5% 6% 4.% Alexa 5% 4% 7% 4) Write the system of inequalities that fits the following graph: y > x 4a) Write the system of inequalities that fits the following graph: 4b) Write the system of inequalities that fits the following graph: 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 (0,5) (4,5) (0,0) (8,0) y < x + y > x P = 4(0) + 8(5) = 40 P = 4(0) + 8(0) = 0 P = 4(8) + 8(0) = P = 4(4) + 8(5) = 56 (4, 5) (0,) (0,0) 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 (,4) (6,0) 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

6 6) Yum s Bakery bakes two breads, A and B. They want to maximize their profits from bread sales. One batch of A yields a profit of $40. One batch of B yields a profit of $0. One batch of A uses 5 pounds of oats and pounds of flour. One batch of B uses pounds of oats and pounds of flour. The company has 80 pounds of oats and 5 pounds of flour available. Write objective quantity and the constraints. Objective Quantity for Maximizing P = 40a + 0b Constraints : x 0 oats and flour are not negative y 0 5x + y 80 constra int for lbs of OATS x + y 5 constra int for lbs of FLOUR } } Profit : 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. Because of space restrictions, the store has room for at most 8 of bookcase A and of bookcase B. Write objective quantity and 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 Objective Quantity: Constraints: Unit : Absolute Value Equations, Inequalities, & Functions 7) Solve for x: 9+ x = 7a) Solve for x: 8 x = 8 7b) Solve for x: 9x + 9 = 6 Positive Case : Negative Case : 9 + x = 9 + x = x = x = x = x = 7, 7 8) Solve & graph: 4x 0 < 0 8a) Solve : x 9 > 8b) Solve: 6 4x -5 0 Case : + Case : 0 < 4x 0 < < 4x < > x > 5 5 < x < 0 ( 5, 0) 9) Given the equation y = x +, determine all transformations. Stretch, Right, Up 9a) Given the equation y = x +, determine all 5 8 transformations. 9b) Given the equation y = x , determine all transformations.

7 0) Given the equation y = x +, determine vertex and whether it s a maximum or minimum. Vertex (, ) " a > 0 " Minimum 0a) Given the equation y = x + 5 8, determine vertex and whether it s a maximum or minimum. 0b) Given the equation y = x , determine vertex and whether it s a maximum or minimum. ) Solve: ) Solve: ) Solve: 9x = x x x x 0 = 0 x 0 x 5 5x x 4 4x x (x + 4)( x 5) = 0 4 {,5 } 4x 8x = 0 4x 8x = 0 4 xx ( ) = 0 { 0, } 9x = x 9x x = 0 (not factorable) x Unit 4: Quadratic Functions a) Solve: a) Solve: a) Solve: x x 6 0 = b) Solve: x + 0x = 0 b) Solve: 4x 4 = x b) Solve: 5x + 9x = 6 x = 4 x x + 6x = x = ± b b 4ac a ( ) 4(9)( ) ± x = (9) ± 80 ± 6 5 ± 5 x = = = 8 8 4) Solve the equation: ( x + ) = ( x + ) = ( x + ) = 4 x + = ± x = ± 0, 4 { } 4a) Solve the equation: 4( x ) = 00 4b) Solve the equation: 9( x + ) = 8

8 5) Simplify: i 6) Find the discriminant of the quadratic equation and then state the number and type of solutions for the equation. x + 4x + = 8x 6 x + 4x + = 8x 6 8x + 6 8x + 6 x + 6x + 9 = 0 5a) Simplify: 7 5b) Simplify: 50 6a) Find the discriminant of the quadratic equation and then state the number and type of solutions for the equation. x + 5x 6 = x 0 6b) Find the discriminant of the quadratic equation and then state the number and type of solutions for the equation. x + 4 = x discri min ant : b 4ac (6) 4()(9) = 0 real solution 7) 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) 7a) The graph of a quadratic equation is shown below. Use the graph to identify the solutions of the equation. 7b) The graph of a quadratic equation is shown below. Use the graph to identify the solutions of the equation. Solutions : {, 4} 8) Describe a in the equation fx () = ax + bx+ cif the graph opens up. a > 0 9) What are the coordinates of the minimum of this function? fx () = x x+ 5 Vertex : x = b = = = a () 4 (, ) 8a) Describe a in the equation fx () = ax + bx+ cif the graph opens down. 9a) What are the coordinates of the minimum of this function? y = x 6x+ 8b) Describe a in the equation fx () = ax + bx+ cif the graph stretches. 9b) What are the coordinates of the maximum of this function? y = x + 8x 5 y = + = () () 5 0) Graph the function fx () = x 4x+ Must find vertex by hand ( middle of table ) Vertex : x = b = 4 = 4 = a () 0a) Graph the function fx = x + 8x+ 6 0b) Graph the function fx () = x 4x+ y = + = x y () 4() (, ) 0 x y x y -

9 Plot the table values to graph ) Write a possible equation (in vertex form) for the given graph: Vertex : (,4) Vertex form : y = x + + ( ) 4 a) Write a possible equation (in vertex form) for the given graph: b) Write a possible equation (in vertex form) for the given graph: ) What are the s of the graph of this function? y = x + x 0 = x + x 0 = ( x + x 6) 0 = ( x + )( x ) x =, a) What are the s of the graph of this function? y = x + 4x + 64 b) What are the x intercepts of the graph of this function? y = x 5x+ 8 x intercepts : (,0),(,0) ) Rewrite the following into vertex form and find the vertex: y = 9x 8x 55 y = 9 x x half ( ) ( x ) Vertex form: y = 9 64 vertex:, 64 square a) Rewrite the following into vertex form and find the vertex: y = x + x b) Rewrite the following into vertex form and find the vertex: y = x 4x+ 0 For # 5: Work down each column to analyze the graph of each function. a) Determine the domain and 4a) Determine the domain and f x = x + 6x + 7 range of: f ( x) = x 8x 9 range of: 5a) Determine the domain and f x = x + 4x range of: : Domain Range : (, ), ) Domain: Range: Domain: Range:

10 b) State the interval over which the function above is increasing and decreasing. Increasing: (-, ) 4b) State the interval over which the function above is increasing and decreasing. Increasing: 5b) State the interval over which the function above is increasing and decreasing. Increasing: Decreasing: (-,-) Decreasing: Decreasing: 6) Given f ( x) = x + 6x + 7, determine the average rate of change for the interval,. f x f x ( ) ( ) + ( ) = fa fb b a f = = = = f = = = 9+ 7 = ( ) ( ) 4 = = = 7) Write the system of equations and determine the solution(s). 6a) Given f ( x) = x 8x 9, determine the average rate of change for the interval 4,. 7a) Write the system of equations and determine the solution(s). 6b) Given f ( x) = x + 4x, determine the average rate of change for the interval,. 7b) Write the system of equations and determine the solution(s). f(x) = l x+l - g(x) = - ( x-) + Solution(s): ( 0, - ) & (, ) 8) Graph the system of inequalities. Shade the appropriate solution set. ( x ) < x f x g x 4 Solid Shade above Solution(s): 8a) Graph the system of inequalities. Shade the appropriate solution set. ( x ) x + f x < g x 5 Solution(s): 8b) Graph the system of inequalities. Shade the appropriate solution set. ( x ) < x > + + f x g x 7

11 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 where x is the time in seconds and f is the height in feet. a) What is the maximum height Jason reaches? Find the y value for the vertex: Vertex : x = b = 6 = a ( 6) ( 4) ( ) y = = = = ft b) How high above the ocean is Jason as he jumps? Find the y intercept "c" : 480 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 a) What is the maximum height of the dud missile? b) How high above the ground is the military instillation? 9b) A rock is thrown upward from the top of a tower with an intial upward velocity of 00 ft/sec. The height of a rock above the ground as a function of time can be modeled by the equation: f ( x) = 6x + 00x + 5 a) What is the maximum height of the rock? 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 Think of: Odd When odd & positive think of +x : When odd & negative think of -x : When even & positive think of +x : When Even & negative think of -x : f ( x) as x, as x, f x Unit 5: Polynomial Functions 40a) Describe the end behavior of the graph of 4 fx = 4x 4x x+. 40b) Describe the end behavior of the graph of 9 4 fx = x + 6x x 5.

12 4) Use the function to determine: 4a) Use the function to determine: 4b) Use the function to determine: 4 = f ( x) = x x + 4x 6 f x x x x = f x x x x (, 4) (., ) Increasing: Decreasing: ( 4,. ) Local Min: y = 448. Local Max: = Positive Interval: 5, , Negative Interval: y 5 ( ) ( ) (, 5) (. 6, 0. 8) 4) Determine if the graph is a function: s Increasing: Decreasing: Local Min: Local Maxima: Positive Interval: Negative Interval: 4a) Determine if the graph is a function: Increasing: Decreasing: Local Min: Positive Interval: Negative Interval: Local Max: 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 x 6 5x 6 x ) Identify the following: 4a) Factor: 4x + 4x 5x 0 44a) Identify the following: 4b) Factor: 4 5 4mn + xm + 8m + 7xn 44b) Identify the following: Degree: Even or Odd Leading Coefficient: Positive or Negative Real Zeros: 4 5 Possible Degree: 4 5 Degree: Even or Odd Leading Coefficient: Positive or Negative Real Zeros: 4 5 Possible Degree: 4 5 Degree: Even or Odd Leading Coefficient: Positive or Negative Real Zeros: 4 5 Possible Degree: 4 5

13 45) Divide: (x 9x 6) (x 9) 7x + 8 x 9 x 9x 6 x 6x 4x 6 4x a) Divide: (8y + y ) (4y ) 45b) Divide: 4 (r 9r + r 6r + ) (r ) 7x x 9 46) Divide : ( n + 6n n 8) ( n+ ) 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 fx = x + x 9x 9, find the other zeros a) If is a zero of fx = 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) Find the remainder when x 8 x + 8 x is divided by ( x + ). Evaluate f(-) 8 f = + 8 = + 8 = 6 The remainder is -6 49) Write the polynomial function of least degree that has zeros of 5, i and i. 48a) Find the remainder when x + x 4 x + 7 is divided by ( x ). 49a) Write the polynomial function of least degree that has zeros of 4, i and i. 48b) Find the remainder when 6 x x 4 x. + is divided by 49b) Write the polynomial function of least degree that has zeros of, i and i.

14 ( x + 5)( x i)( x + i) = f ( x ) ( 5)( 4 ) x + x + xi xi i = f ( 5)( 4 ) x + x i = f ( x ) ( ( ) ) ( x ) ( x + 5) x 4 = f ( x ) ( x + 5) x + 4 = f x + 4x + 5x + 0 = f ( x ) ( x ) fx x x x = ) If fx () = x + 5x 4 and gx () x 5 = +, find fx + 4 gx. 4 50a) If fx () = 4x x + and gx x =, find fx () 4 gx (). 6 50b) If fx () = x x + and gx 4x 6 = +, find () fx+ () gx. ( x + 5x 4) + 4( x + 5) x 5x + 4 4x + 0 x 9x + 4