MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the coordinates of the midpoint of AB. Can ou find the - and -intercepts of the graph of an equation algebraicall? (. &.) () Find all - and -intercepts of the graph of each equation. (a) + = (b) = 9 Can ou graph a line given the equation of the line or write the equation of a line given information about the line? (.) () Write an equation of the line passing through (, 9) and perpendicular to the line Epress our answer in standard form and in slope-intercept form. + =. Can ou use a graphing utilit to find the line of best fit to a data set, and can ou interpret the slope of the line of best fit? (.) () The table shows Test scores vs. final course averages for a particular class. Use a graphing utilit to find the line of best fit to the data. Interpret the slope of the line of best fit. Test Score Final Course Average 70 79 78 89 67 76 6 79 7 68 86 97 77 8 8 9 6 79 Can ou solve an inequalit algebraicall? (.) () Solve the inequalit 9 <. Epress the solution in interval notation, and graph the solution set. Revised April 0
Can ou determine whether the graph of an equation will be smmetric with respect to the -ais, - ais, or origin algebraicall? (.) (6) Determine whether the graph of 9 = is smmetric with respect to the -ais, -ais, or origin. Can ou graph a function using transformations? (. &.) (7) Given here is a complete graph of a function f. Use this graph to sketch the graph of F ( ) = f ( + ). 6 6 Can ou solve an equation or inequalit involving absolute value algebraicall? (.) (8) Solve = 7. (9) Solve. Epress the solution in interval notation. Can ou graph, evaluate, and determine characteristics of a piecewise-defined function? (.) (0) Consider the function f ( ) (a) State the domain of f. (b) Identif an intercepts of f. = if if <. (c) (d) Sketch a graph of f b hand. Find the range of f using our graph. Can ou graph transformations of a power function b hand? (.,., &.) () Graph ( ) = ( +) Revised April 0 f using transformations.
Can ou find the sum, difference, product, quotient, and composition of two functions, and can ou state the domains of each? (.6) () Find the composite function ( f o g) ( ) given f ( ) of ( f o g) ( ). = + and g ( ) =. State the domain Can ou perform operations on comple numbers and find the comple solutions of a quadratic equation? (.) () Write + i i in the standard form a + bi. () Solve the equation + + = 0 in the comple number sstem. Can ou graph a quadratic function b hand and algebraicall determine properties of the graph? (.) () Graph ( ) f = 7 b hand. Identif the verte, the ais of smmetr, and an intercepts. Can ou find the real solutions of a quadratic equation algebraicall? (.) (6) Find the real solutions, if an, of = b completing the square. Can ou solve an equation that is quadratic in form algebraicall? (.) (7) Find the real solutions, if an, of + + 6 = 0. + + Can ou determine characteristics of a function, including domain and range, average rate of change, intervals of increase and decrease, local maima and minima, and whether the function is odd or even? (.) (8) Consider ( ) f =. (a) Find the range of f. (b) Find the local maima or minima of f. (c) Determine algebraicall whether f is odd, even, or neither. Can ou analze a polnomial function and its graph? (.) (9) Consider the polnomial function f ( ) ( ) ( ) = +. (a) Algebraicall find the - and -intercepts of f. (b) Determine for each - intercept whether the multiplicit is even or odd, and thus determine whether the graph of f will touch the -ais or cross the -ais there. Revised April 0
(c) Draw an end behavior diagram for f. (d) Confirm (a) (c) using a graphing utilit. (e) Determine the number of turning points on the graph of f. (f) Determine an local maima and local minima of f. Can ou find the real zeros of a polnomial function? (.7) (0) Use the Factor Theorem to determine whether is a factor + 8. If it is, write f in factored form. Can ou find the comple zeros of a polnomial? (.7) () Find the comple zeros of + 60. Can ou use a graphing utilit to solve an equation? (.8) () Use a graphing utilit to approimate the real solutions, if an, of the equation rounded to two decimal places. = π + Can ou algebraicall find the domain of a function, identif the graph of a function, and obtain information from or about the graph of a function? (.) () Consider f ( ) =. + (a) Is the point (,8) on the graph of f? (b) If = 0, what is f ( )? What point is on the graph of f? (c) If f ( ) = 8, what is? What point is on the graph of f? (d) What is the domain of f? Can ou solve a radical equation algebraicall? (.) () Find the real solutions, if an, of + = +. Can ou solve polnomial and rational inequalities? (.) () Solve the rational inequalit + + algebraicall. Can ou graph a circle b hand given the equation of the circle? (6.) (6) Find the center ( h, k ) and radius r of the circle + + 6 = 0. Graph the circle b hand. Revised April 0
Can ou write the equation of a circle given information about the circle? (6.) (7) Find the standard form of the equation of the circle with endpoints of a diameter at (, ) and (,7 ). Can ou graph a parabola given the equation of the parabola or write the equation of a parabola given information about the parabola? (6.) (8) A parabola has verte (, ) and directri form and graph the parabola b hand. =. Write the equation of the parabola in standard Can ou graph an ellipse given the equation of the ellipse or write the equation of an ellipse given information about the ellipse? (6.) (9) An ellipse has equation ( ) ( ) + + =. Identif the coordinates of the center, the vertices, 6 and the foci of this ellipse, and graph the ellipse b hand. Can ou graph a hperbola given the equation of the hperbola or write the equation of a hperbola given information about the hperbola? (6.) (0) A hperbola has center (, ), a verte at ( 0, ), and a focus at (, ). Write the equation of the hperbola in standard form. Can ou solve sstems of linear equations, including sstems that are inconsistent and sstems that consist of dependent equations? Can ou use a sstem of linear equations to model a problem mathematicall? (7. & 7.) () Solve the given sstem using substitution or elimination. + = = 9 7 () Solve the given sstem b elimination. z = + z = 7z = 0 () Solve the given sstem. + z = 0 + z = + 6 + z = Revised April 0
() Write a sstem of equations to model this problem, and then solve algebraicall. Bob received a lotter check for $0,000. He invested the mone into three different accounts. Part of his mone was placed in a savings account paing 7% interest. A second portion, which was twice the first amount, was deposited in a CD paing 9% interest. The remainder of the mone was put into a mone market fund ielding 0% interest. If Bob s total interest over a one-ear period was $9, how much was deposited in each of the three accounts? Can ou set up and solve a sstem of nonlinear equations? (7.) () A rectangle has area 0 in and perimeter 6 in. Find the length of a diagonal of the rectangle. Can ou find the n th term of a sequence and epress a sum using summation (sigma) notation? (8.) (6) Write down the apparent n th term of the sequence,,, 7, 9,.... (7) Epress the sum + ( + ) + ( + ) + ( + 9) + ( + 6 ) +... + ( + ) in summation notation. Can ou find a formula for the n th term of an arithmetic sequence, and can ou solve an application problem involving the sum of the terms of an arithmetic sequence? (8.) (8) The rd term of an arithmetic sequence is 9, and the 8 th term of the same sequence is. Find a formula for the n th term of the sequence. (9) A trapezoidal seating area contains 0 seats in the first row. Each successive row contains fewer seats than the previous row. The last row contains 0 seats. How man seats are there in this seating area? Can ou find a formula for the n th term of a geometric sequence, and can ou find the sum of a geometric series? (8.) (0) Find an epression for the nd term of the sequence 6 8,, 8,,.... () Find the sum if it eists: 6 8 + + 8 + +.... Can ou epand a binomial using the Binomial Theorem? (8.) () Epand ( ) using the Binomial Theorem. Revised April 0
Answers (with occasional eplanations): () (a) 0 (b), () (a) -intercept, 0 ; -intercepts ( 0, ± ) (b) -intercepts( ±, 0) ; no -intercept () = 0 (standard form); = 0 (slope-intercept form) () 0.7 + 9.88. For ever one-point increase in the Test score there is, on average, approimatel a 0.7-point increase in the final course average. () ( 7, ] ( ] 8 7 6 0 6 7 8 (6) Onl replacing both b and b ields an equivalent equation, so the graph is onl smmetric with respect to the origin. (7) 6 6 (8) {, 6} 7 7 (9),, Revised April 0
(0) (a) (,] (b) no -intercept; -intercept is (c) 6 6 (d) [ ),0 [, ) () Start with the graph of =. Shift the graph to the left unit, reflect it over the -ais, and then shift it up units. 6 6 () ( f o g) ( ) =. The domain is (,] () Multipling b Revised April 0 + + i + i ields 8 9 + i..
() = i, + i 7 () The verte is,. The ais of smmetr is =. The -intercepts are 7 = ± or {.,.}. The -intercept is 7. (6) 7 7 =, + (7) = {, } (8) (a) [, ) (b) Local Min at = or located at the point (,). (c) f ( ) = ( ) ( ) = +, which is neither f ( ) nor f ( ). So f is neither odd nor even. (9) (a) -intercepts: and ; -intercept: 6 (b) is of even multiplicit, so the graph will touch the -ais there. is of odd multiplicit, so the graph will cross the -ais there. (c) (d) 0 0 0 - - -0 (e) turning points (f) local minimum 0 at = ; local maimum at = Revised April 0
(0) f ( ) = 0, so is a factor of f. The factored form of f is ( ) ( + ). () = {,, i, + i} () = {.,.9} () (a) no (b) If = 0, f ( ) = 0. So, the point ( 0,0 ) is on the graph of f. (c) If f ( ) = 8, = 6. So, the point ( 6,8 ) is on the graph of f. (d) (, ) () = 6 () (, ) [,) 9 (6) Center = (,) radius = 0.6 7 6 6 7 6 (7) ( ) 7 + = (8) ( ) = 6( ) 9 8 7 6 6 7 8 9 Revised April 0
(9) Center (,). Vertices (, ) and (,6). Foci (, ) 6 ±. 7 6 6 7 6 (0) () ( ) ( + ) = 7, 8 () There are infinitel man solutions, each of the form ( ) () (,, ) + z, + z, z, z. () Let s represent the amount invested in savings. Let c represent the amount invested in the CD. Let m represent the amount invested in the mone market fund. The sstem we initiall write is s + c + m = c = s 0000 0.07s + 0.09c + 0.0m = 9. After solving, we find that Bob has invested $00 in the savings account, $000 in the CD, and $00 in the mone market fund. () Letting l be the length of the rectangle and w be the width we have lw = 0 and l + w = 6. Substituting 0 0 for w in the second equation gives us l + = 6. We multipl through b l l l and solve the resulting quadratic equation, giving us a length of in. (and a width of 8 in.). The diagonal thus has length 8 + in, or 7 in. Revised April 0
n (6) a = ( ) ( n ) n (7) ( + k ) k= 0 (8) ( ) a = 7 + n = n (9) 60 seats n (0) a = 8 a 8 () S = = = r () Before simplifing, we get 0 + + + + Simplifing, this becomes 8 0 + 0 00 + 6. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 Revised April 0