1010 REAL Review for Final Exam

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1 1010 REAL Review for Final Exam Chapter 1: Function Sense 1) The notation T(c) represents the amount of tuition paid depending on the number of credit hours for which a student is registered. Interpret the meaning of T(12). (Section 1.2) 2) You know the point (2, ) is on the graph of a function f. Give the coordinates of a point you know is not on the graph of the function. ) Use what you ve learned about solving systems of equations (Section 1.12) and solving inequalities (Section 1.15) to determine if the ordered pair (, -5) is a solution to this x + 7y > 1 system of linear inequalities {? Explain your answer. 2x 5y < 4 4) For every function involving two variables, we designate one variable as input and the other variable as output. (Section 1.2,1.) a) Which of these is always the dependent variable? b) On which axis (horizontal or vertical) is the dependent variable always labeled? 5) The amount of water in a leaky bucket is given by the linear function y = 126 4x, where y is in ounces and x is in minutes. Find and interpret the slope and y-intercept of the linear equation. (Section 1.7) 6) In words, give the definition of slope and tell how slope and average rate of change are related. 7) The price of a certain commodity is a function of supply and demand. The table below shows the price of the commodity per barrel between 2005 and (Section 1.8) a) Find the average annual rate of change between 2006 and b) In terms of average rate of change, explain why this is or isn t a linear function. Year Price/barrel 2005 $ $ $ $ $ $6 8) a) Graph the function f(x) = -x + 8 as a solid line. b) Find the equation of its inverse and write it in function form. c) Graph the inverse as a dashed line on the same set of axes.

2 d) Given that the domain and range are switched for inverse functions, how could you graph the inverse function without first finding the equation of the inverse? (Sections 1.7, 2.8) 9) Solve the following system of equations. If the system is dependent or inconsistent indicate this. If not, give the solution as an ordered triplet. (Section 1.1) x + y + z = -2 x - y + 4z = -8 4x + y + z = 1 10) Solve the inequality, giving its solution set in both interval and graph forms. (Section 1.15) -4(y + 6) < -16y + 12 Chapter 2: The Algebra of Functions/Radicals 11) Is it correct to simplify 2 2 as 6 6? Explain why or why not. (Review of Exponents) 12) Expand (x 4). 1) Divide x 5x x by x ) Can a represent a negative real number? Why or why not? 15) Does the equation x + 2 = x have a real-number solution? Explain. 16) A student tries to solve x + 4 = x + 5 by squaring both sides and gets x + 4 = x Why is this not correct? 17) Solve the equations algebraically. Round to three decimal places when necessary. Check your answers. (Section 5.8) a) (x 2) 1 2 = 6 b) 4 x = 9x + 9 c) 2x + 15 x = 6 18) Explain how to evaluate ( 125) 4. 19) Simplify the expression involving rational exponents without using a calculator. Show each step. (Section 2.4) a) b) ( 8) 2 c) ( 49 1 ) 2 81 d) ( 8 4 ) 27

3 20) Use the rules of exponents to simplify the expression. Write the answer with positive exponents. Assume that all variables represent positive real numbers. 1 a) ( x4 4 y 8) b) (24k 5 m 10 ) ) Express either as a simplified radical or rational number. Assume that all variables represent positive real numbers. a) 245k 7 q 8 b) 84 c) 27 22) Rationalize the denominator a) 18x x 5 b) x5 2 c) ) Explain why the expression a 11 a cannot be simplified further ) Can the sum 50, ,005 be simplified? What would be required to do so? 25) Perform the indicated operations and simplify. Assume that all variables represent positive real numbers. a) b) ) Determine whether the statement x correct answer. Explain. x = x is true or false. If it is false, give the 27) Find the requested composition of functions. (Sections 2.5, 2.6/S1) a) Given f(x) = 4x 2 + x + 6 and g(x) = x - 5, determine g(f(x)). b) Given f(x) = -5x - 7 and g(x) = -x 2 + 6x - 9, determine g(f(-4)). 28) You drop a pebble off a bridge. Ripples move out from the point of impact as concentric circles. The radius (in feet) of the outer ripple is given by R = f(t) = 0.5t where t is the number of seconds after the pebble hits the water. The area, A, of a circle is a function of its radius and is given by A = g(r) = πr 2 a) Determine a formula for g(f(t)) b) What are the input and output for the function defined in part a?

4 Chapter : Exponential and Logarithmic Functions 29) The smoking prevalence among adults as a percent of the US population can be modeled by the function p(t) = 7.6(0.984) t where t equals the number of years since (Section./S14) a) From the equation, determine the vertical intercept. What practical meaning does this intercept have in this situation? b) Determine the growth or decay factor? c) If this exponential trend continues, what will be the approximate percent of US adults who will smoke in the year 2020? Round to the nearest percent. 0) Your starting salary for a new job is $25,000 per year. You are offered two options for salary increases: Plan 1: an annual increase of $1,000 per year or Plan 2: an annual increase of % per year (This would mean an exponential growth factor of 1.0) a) Your salary is a function of the number of years of employment at your job. Write an equation to determine the salary, S, after x years on the job using plan 1 and another equation to determine your salary using plan 2. (Sections 1.8 and.1-./s14) 1) For the following data sets determine: a) Is it linear or exponential? b) Does it have a slope, growth factor, or decay factor AND what is that value? c) Is the function increasing or decreasing? d) Write the equation for the function. Data Set #1 X Y

5 Data Set #2 X Y ) You are planning on purchasing a new car and have your eye on a specific model. You know that new car prices are projected to increase at a rate of 5% per year for at least the next five years. a) Write an equation that represents the projected cost C of your dream car t years in the future, given that it costs $27,000 today. (Use the following table of the first few years to figure the growth factor if needed.) b) Use your equation to project the cost of your car 4 years from now. (Section./S14) t C ) Graph the function f(x) = 4 x and label three points (using ordered pairs). (Section.1) a) What is the domain of the function? b) What is the range of the function? c) For what values of x is the graph increasing? d) What is the x-intercept? e) What is the y-intercept? f) Does the graph have a horizontal or vertical asymptote? What is it? 4) Without graphing, classify the function f(x) = 8 ( 1 2 )x as increasing or decreasing, and determine f(0). (Section.2)

6 5) Graph the function f(x) = ( 1 2 )x and label three points (using ordered pairs). (Section.2) a) What is the domain of the function? b) What is the range of the function? c) For what values of x is the graph increasing? d) What is the x-intercept? e) What is the y-intercept? f) Does the graph have a horizontal or vertical asymptote? What is it? 6) Why can't y = 2 x have an x-intercept? (Section.1) 7) Complete the following table representing an exponential function. Round calculations to two decimal places if necessary. (Sections.1-./S14) x y ) For each exponential equation, write an equivalent equation in log form. (Section.8/S15) a) 6 = 216 b) 2 = 1 9 c) 10 2 = ) When asked to describe logarithms in one simple sentence, the teacher said, "Logarithms are exponents". What is meant by this sentence? (Section.8/S15) 40) One of your classmates says, a logarithm is an inverse? Is this true? 41) For each log equation, write an equivalent equation in exponential form. (Section.8/S15) a) log = b) log = 5 c) log 9 = 2 42) Solve the equation for the unknown variable. (Section.8) a) log 2 8 = x b) log 6 = a c) log 9 1 = n d) log 7 (7 9 ) = x e) log e 5 = lnx f) ln e = x d) ln e 7 = x e) log 5 5 = x 4) Graph the function f(x) = log x and label three points (using ordered pairs). (Section.9) a) What is the domain of the function? b) What is the range of the function? c) For what values of x is the graph increasing? d) What is the x-intercept?

7 e) What is the y-intercept? f) Does the graph have a horizontal or vertical asymptote? What is it? 44) Graph the function f(x) = log1 x and label three points (using ordered pairs). (Section.9) a) What is the domain of the function? b) What is the range of the function? c) For what values of x is the graph increasing? d) What is the x-intercept? e) What is the y-intercept? f) Does the graph have a horizontal or vertical asymptote? What is it? 45) Determine the equation of the inverse of each function. How is this process different than just changing the form of the equation from log to exponential form or from exponential to log form? (Section.9) a) y = 8 x b) y = log 9 x Chapter 4: Quadratic Functions 46) Explain in words why we can solve quadratic equations by setting them equal to zero and factoring them. 47) Solve the following quadratic equations by factoring. (Section 4.4/S16/S2) a) 6x 2 7x = 0 b) x 2 2 = x c) x 2 12x = 0 d) x x 4 = 0 48) Use the square root property to solve the equation (x 9) 2 = 64 49) An arrow is shot vertically into the air from a height of 5 feet with an initial velocity of 96 feet per second. The height, h, in feet above the ground, at any time, t (in seconds), modeled by h = t 16t 2. Answer the following questions accurately to two decimal places. (Section 4.2) a) In how many seconds will the arrow hit the ground? b) What is the maximum height that the arrow will reach? c) When will it reach a height of 85ft? d) What is the practical range for this function?

8 50) Solve the equation z z + 18 = 0 by completing the square. (Supplemental Activity) 51) Use the quadratic formula to solve the equation 5n 2 = 12n 5. (Section 4.5) 52) List four methods used to solve quadratic equations. Which of the four methods of solving quadratic equations work(s) for any quadratic equation? (Sections 4.4/S22/S2-4.5) 5) Which one of the following methods cannot be used to solve the equation x 2 4x 6 = 0? (Sections 4.4/S22/S2-4.5) A) Completing the square B) All of the methods can be used. C) Quadratic formula D) Factoring 54) The vertex of a parabola is located at the point ( 1, 1). The point ( 4, 5) also lies on the parabola. Determine the coordinates of a third point that MUST lie on this graph. 55) The x-intercepts of a parabola are ( 5, 0) and (11, 0). Determine the equation of its axis of symmetry. 56) Graph the function f(x) = 2x 2 4x + 5. State the axis of symmetry. Label (using ordered pairs) the vertex and two additional points. (Section ) 57) Write the ordered pairs of the x- and y-intercepts of the graph of f(x) = x 2 6x 7. (Section 4.1,4.2) 58) Find the complex solutions of the equation x 2 2x + 7 = 0. Write in a+bi form. (Section 4.7/S24) 59) We apply the quadratic formula and find that the value of b 2 4ac equals zero. (Section 4.7) a) What does this tell us about the number of solutions to the quadratic equation? b) What does this tell us about the number of x-intercepts on the graph of this quadratic function? c) Sketch what the graph of the parabola might look like on a Cartesian coordinate plane. 60) The given graph is f(x). Draw the graph of f(x) 1 on the same set of axes.

9 61) Add or subtract as indicated. Write your answers in standard form. (Section 4.7/S24) a) (5 + 4i) - (-4 + i) b) [(10 + 8i) - (6 + 6i)] - (2-10i) c) (8-4i) + (5 + 2i) 62) Multiply (7-5i)(2-5i). (Section 4.7/S24) Chapter 5: Rational Expressions and Equations 6) Does the rational expression 5x+7 x+7 reduce to 5? Why or why not? 64) If the rational expression 5x9 y 6 4p 9 q represents the area of a rectangle and 4x8 y 4 p 8 the length of the rectangle, what rational expression represents the width? represents 65) What are two possible LCDs which could be used for the sum 6 x x? 66) If one form of the correct answer of a difference of two rational expressions is k 6, what would be an alternate form of the answer if the denominator is 6 - k? 67) Simplify the complex fractions. (Section 5.6/S20) a) b) c) 4+ 2 x x d) 4 x x ) Explain the difference in the procedures used for adding rational expressions and solving rational equations. (Sections 5.5, 5.6/Lab 6) 69) If (7x - 6)(4x - ) is the LCD of two fractions, is (6-7x)( - 4x) also acceptable as an LCD? Why or why not? 70) Solve the equations algebraically. Round the answer to three decimal places whenever necessary. (Section 5.5/S19) a) 10.86x 14 = 14. b) = x x x c) 5 x = 18

10 71) Multiply or divide as indicated. Write the answer in lowest terms. (Section 5.6/S17) a) 4r 12 5r+10 10r 2 +20r 24 8r b) y y 20 y+4 y+2 c) 2t 2 t 9 t2 +12t 6 t 2 t 6 t 2 +t 18 d) z 2 +10z+25 z2 +5z z 2 +1z+40 z 2 +6z 16 72) Add or subtract. Write the answer in lowest terms. (Section 5.6) a) x 4 x 2 16 x 2 +5x+4 b) 4m m m m 2 m 2 10m c) y y 5 7) Suppose that the concentration of a particular drug in the bloodstream, measured in milligrams per liter, can be modeled by the function C = 15t t 2, where t is the number of +2.5 minutes after injection of the drug. What is the concentration of this drug 4 minutes after the injection? Round the answer to three decimal places whenever necessary. (Section 5.6) 74) A formula for electric circuits is 1 R = 1 R R 2. If R 1 = 10 ohms and R 2 = 1 ohms, find R. Round the answer to three decimal places. (Section 5.6/S19) 75) Solve for T in the formula 1 t t 2 = 1 T (Section 5.5/S19) 76) Complete the following table by writing an example of an equation that fits each description and sketching a graph that would fit each description. Label at least two points for each graph. Description Equation Sketch Graph Constant Function Linearly Decreasing Function Logarithmically Increasing Function

11 Exponentially Increasing Function Logarithmically Decreasing Function Exponentially Decreasing Function Quadratic Function with Complex Zeros No need to label points, just sketch. After completing a test you should verify that you did the following things correctly: a) Distribute negative signs correctly over parentheses b) Get three terms after squaring a binomial. (x+2) = x 2 + 4x + 4, not x c) Throw out extraneous solutions d) Did not cancel terms, but divided out ones. Remember you can only divide what is being multiplied (aka factors)