Math Final Eam Review Problems Rockville Campus Fall. Define a. relation b. function. For each graph below, eplain why it is or is not a function. a. b. c. d.. Given + y = a. Find the -intercept. b. Find the y-intercept. c. Find the slope. d. Sketch the graph. e. Find the equation of a line parallel to this line passing through (-,-). Given ( ) f = a. Find the verte f ( ) of by completing the square. b. Find the verte of f ( ) by using the formula for the -coordinate of the verte. c. Use your graphing calculator to find the -intercepts of f ( ) d. What is the y-intercept of f ( )?. Simplify a. b.. Let f ( ) = a. Find and simplify f ( ) b. Find when f ( ) = c. +
. Given ( ) f = + g( ) = h( ) = For each function above, use any suitable method to answer the following questions: a. Find all intercepts. When it is required to estimate them, estimate to the appropriate number of digits. b. Evaluate the functions at = and =. c. Sketch a graph that shows the significant features of the function. d. If the function is a parabola, find the eact value of the -coordinate of the verte.. Simplify each epression below where possible and write your answer without using negative or fractional eponents. Assume that >. a. e. b. ( ) c. ( ) d. ( )( ). Simplify and epress the answer in a + bi form.. Simplify a. b.. Find all real and non-real solutions to the following: a. ( + ) = b. + = d. + = c. ( + )( ) =. A company started business in. Assume that a company has a profit per year given by the equation f () t =.t +.t. where f ( t ) represents the amount of profit in millions of dollars, where t is the number of years since. a. Use f to predict the amount of profit in. b. Find t when f () t =.. What do your results mean in terms of the company s profits. c. For what years is there likely model breakdown? (Hint: consider the past and future.) d. When does f () t =? What does this mean in terms of the situation? e. When will the company have maimum profits?. Your calculator gives an answer of =., y = E- when using ROOT or ZERO in the graphing mode. Give the coordinates of the -intercept your calculator has given you to the nearest thousandth.. Solve for algebraically. a. + = + + b. + =
. The equation of the height of an object thrown upwards starting at feet above the ground is h() t = + t t where t is time in seconds and h is height in feet. a. When will the object hit the ground? b. When will it reach its maimum height? c. What is its maimum height? + = Evaluate each of the following. If you cannot evaluate, eplain why.. Let f ( ) f ( ), f ( ), f ( ), f (), f ( ), f ( ). What is the domain of f if f ( ) = +?. Bacteria are grown in a dish. The chart above is a graph of the number of bacteria in the dish. Food is added as needed. The vertical ais is the number of bacteria in thousands. The horizontal ais is the time in hours. a. Approimately how many bacteria are in the dish after five hours? b. After hours the number of bacteria appears to be leveling off at what number? Eplain why this might be the case.. Evaluate a. () log b. ( ) ln e c. ( ) ln d. log. Which part of the previous problem is difficult to do without a calculator? Eplain.. Label all intercepts and sketch the graph of a. f ( ) = f = ln b. ( ) ( ) c. f ( ) =
. Function I Function II - - - - - - - - - - - - - - - - - - - - Function III - - - - - - - - - - - - - - - - - - - - Function IV - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Refer to the charts above to answer the following. a. List all the functions for which f ( ) =. b. List all the functions for which f ( ) =. c. What are the zeros of function III? d. Approimate the equation of the line in function I. $ is deposited in a savings account that pays % interest compounded annually. a. How much is in the account after years? b. Use logarithms to determine how many years it will take for the account to grow to y = A t in an appropriate window. $. Confirm your results by graphing ( ). Sketch the graph of a line with the following characteristics: a. m > and b < b. m < and b < c. m = and b >. Solve the following systems of equations algebraically. s t = + y = a. b. s + t = y =. Solve = algebraically.
. A retired man looks at the his net worth since he retired. The data is in the table below. Years since retirement Net worth in thousands a. Create a scattergram of the data. b. Sketch a line that approimates the data. c. Find the equation of that line. d. Let y = f ( ) be the function that models the data. Find f ( ) e. What is the -intercept? What does it mean in terms of the model? f. What is the y-intercept? What does it mean in terms of the model? g. What is the slope of the line. What does it mean in terms of the model?. Use a graphing calculator to solve the following systems. Give your answers to the appropriate number of digits. y = y = + a. b. + y = ( y+ ) = +. Find the domain of each function below. Give your answer using both set builder notation and interval notation. Which of these functions is a polynomial? a. f ( ) = ln ( ) b. g( ) = c. k( ) =. a. Convert = to logarithmic notation. b. Convert log = to eponential notation.. Find the equation for the eponential function of the form points (, ) and ( ),. y = ab that passes through the. Given f ( ) = a + c. a. How does the sign of a affect the graph of f? b. How does the sign of c affect the graph of f?
. a. If f ( ) = find f ( ) b. If g( ) = find g ( ) c. Below is the graph of h( ). Approimate h ( ) - - - - - - - - - - - - - - -. Four tables are given below; three are functions, and one is not. One of the functions is linear, and one is eponential. a. Which one is the linear function? What is the slope of the line? b. Which one is the eponential function? Eplain how you recognize it. c. Which one is not a function? Eplain how you know this relation isn't a function. d. Make a table of the inverse function of the relation in table B e. State the domain and range of the function found in part d. Table A Table B Table C Table D y y y y - - - - - -. Combine and simplify a. + b. +. Solve the following by factoring. a. = b. + = c. + =. Solve by completing the square. a. = = b. ( ) ( ). Solve the following symbolically. a. log log + = b. ( ) e = c. = d. log =
Answers. Note: answers marked by an asterisk (*) are difficult to find without a graphing calculator. ab, and. a. A relation is a set of ordered pairs. b. A function is a relation in such that if ( ) ( ac, ) are ordered pairs in the function, then b= c.. a and d are functions because it is impossible to draw a vertical line that intersects the graph more than once. b and c are not functions because in each case, it is possible to draw a vertical line that intersects both these graphs twice.. a. (,) b. (,) c. d. See the graph to the right. e. y =. a. and b. ( ), *c.., -. d. -. a. b.. a. c. /. c. f g h *a (,), (.,), (,), (.,),,., (.,) b f ( ) =, f ( ) = g ( ) = h ( ) =, g( ) = h = ( ) ( ) d. NA ( ) - - - - Graphs: ( ) f = + g( ) = ( ) h = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -. a. b. c.. i d. e.
. a. b.. a., - b., - c., - d. ± i.. a.. million b. or about.. The company will make a profit of. million in and which is close to what it did in. c.when t is negative the company did not eist. Also the model predicts losses after. d., about. The company would break even then. e. About. (.,). a., (- is etraneous) b., (- is etraneous). a. seconds b. / seconds c. / feet. f ( ) is undefined because is undefined on the reals, f ( ) =, ( ). f () is undefined because division by zero is undefined, f ( )., f ( ) =... a. b.. They ve run out of space.. a. b. c.. d.. c. The others can be evaluated using the properties of logarithms.. a. b. f, c. (,) - - - - - - (,) - - - - - - (,) - - - - - - - -. a. III and IV b. II and IV c.,, d. y =.+. a. $. b.. years
. a. b. c.. a. (, ) b. (,), (, ) ln ( ).., or ln ( ), (, ), (, ) a. & b Graph to the right. c. y =.+ (Other answers are possible.) d.. e.. or about years. He will run out of money in. years. f.. He started with a net worth of $, when he retired. g. -.. His net worth decreased by $, per year.. a. (.,.) b. (.,.) and (.,. ). a. { > } or (, ) b. { and ± } or (, ) (,) (, ) (, ) c. { } or (, ]. f ( ) = + is a polynomial.. a. log = b. =. y = ( ). a. If a > the parabola opens upwards. If a < the parabola opens downwards. b. If c >, the verte and y-intercept are above the -ais. If c <, the verte and y-intercept are below the -ais. f = +. a. ( ) b. ( ) g = log c.. or.. a. The linear function is Table B. The slope of the line is /. b. Table D is the eponential function. For each -unit increase in, the y-coordinate increases by a multiplicative factor of. c. Table C is not a function because there are two output values (y) for same input number (). e. Domain: {-,-,,} Range: {-,,,} + b. +. a., ± b., c.. a. ( + )( + ). a., b.. a. b. ln ± c., d. d. Table B y - - -