( ) ( 4) ( ) ( ) Final Exam: Lessons 1 11 Final Exam solutions ( )

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1 Show all of your work in order to receive full credit. Attach graph paper for your graphs.. Evaluate the following epressions. a) b) ( 0) if ( ) ( ) ( ) 0 0 ( 8 0) ( 4 0) ( 4) 6 c) + 6 if ( ) Duke University Talent Identification Program Page of

2 d) Perform the indicated operations and simplify. Final answers should contain only positive eponents. a) ( + + ) ( ) b) ( ) ( ) ( ) Duke University Talent Identification Program Page of

3 ( 4 c) ) y z 7 y ( 4 ) 4 ( ) y z y z 7 7 y y 6 y z 7 y 6 6 y z 7 y z 4 y y 4 y 7 y z d) e) ( 4 + ) ( 6 + 6) ( ) ( ) f) ( + ) ( 7) ( ) ( ) Duke University Talent Identification Program Page of

4 . Simplify. a) b) 4 ( + 6 ) ( ) c) ( + ) ( + ) + + ( ) d) Duke University Talent Identification Program Page 4 of

5 7 e) ( 0 ) ( 0 ) ( ) ( ) f) ( ) ( ) + ( + ) ( + ) ( ) ( 7) ( + ) ( ) ( 7) ( + ) ( + ) ( ) 4. Solve the following equations for. Please give eact answers, not decimal approimations. a) b) ( ) 009 Duke University Talent Identification Program Page of

6 c) a + b d c a + b d c a + b c c d c a + b cd a cd b cd b a d) 0 0 ( ) ( ) or or 7 or e) 4 6 ( ) ( ) ( ) or + 0 or 0 0 or or 009 Duke University Talent Identification Program Page 6 of

7 . Solve the following equations for. Please give eact answers, not decimal approimations. a) This won t factor, so we have to use the quadratic equation ( 7) ( 7) 4 ( ) ( ) ( ) ± 7 ± ± 7 6 b) ( + ) ( + ) Duke University Talent Identification Program Page 7 of

8 c) ( ) ( ) ( + ) ( ) LCD: ( + ) ( ) ( ) ( ) ( ) ( ) Check: ( ) ( ) ( ) ( ) ( ) or 0 0 or 0 or ( ) ( ) + undefined! Check: Duke University Talent Identification Program Page 8 of

9 d) By definition of principal square root, this has no solution. The principal square root of a number must be positive. Since this says that the square root is -, it has no solution. e) ( ) ( + 0 ) ( ) ( ) or or See check below. Answer: Duke University Talent Identification Program Page 9 of

10 Check: Solve. Graph the solution on a number line Graph the following equations and inequalities. a) y y y + y y-intercept is ( 0, ) ; slope is 009 Duke University Talent Identification Program Page 0 of

11 b) y < 6 y y < 6 < + 6 y-intercept is ( 0,6 ) ; slope is g + + c) ( ) Verte: b a ( ) y g ( ) ( ) ( ) Verte is (,4 ) -intercepts: 009 Duke University Talent Identification Program Page of

12 ( ) ( ) or + 0 or -intercepts are (,0 ) and (,0 ) y-intercept: ( ) ( ) g y-intercept is ( 0, ) Since the square term is negative, the graph opens down. 009 Duke University Talent Identification Program Page of

13 009 Duke University Talent Identification Program Page of 8. Compare and contrast the following: Solutions of a quadratic equation, factors of a quadratic epression, zeros of a function, and - intercepts of a graph of a quadratic function. The solutions of a quadratic equation are the values that make the equation equal to 0. When we factor a quadratic epression, we get two monomials, which when set equal to zero and solved, are the solutions to the equation. The zeros of a function are the values in the domain that make the function zero. And the - intercepts of the graph of a function are the points on the graph where the y-value is 0. This is the same as the zeros of the function (since it is where y f() 0). So, if we know the factors of a quadratic epression, then we know that if that epression were equal to zero, we could use the factors to find the solutions. And these solutions would correspond to the zeros of the function as well as the -intercepts of the graph of the function. 9. If f ( 0) 4 and f ( 4) 6, find an equation for the linear function ( ) Since f ( 0) 4 and ( ) are ( 0,4 ) and ( ) equation of the line. The slope of the line is: 6 4 m f. f 4 6, we know that two points on the line 4,6. We can use these two points to determine the We can find the y-intercept using the slope-intercept formula: y m + b 4 ( 0) + b b 4 b

14 The slope is m and the y-intercept is b 4. The equation of the linear function is: f ( ) Solve the following problem: Two cars left a rest area at the same time. The first car arrived at the net rest area 0 minutes later and averaged a speed of 6 miles per hour. The second car arrived 0 minutes later than the first. Find the average speed of the second car. The formula for distance is D r t. Since the cars traveled the same distance, we can find the formula for the distances that each car traveled and set those distances equal to each other. The first car traveled at a rate of 6 mph and at a time of 0 minutes. In order to keep our units accurate, we ll convert his time to hours. He traveled for 0 60, or, hours. The distance the first 6 car traveled is D r t 6. 6 The second car traveled at a rate of r mph and at a time of 0 + 0, or 60 minutes. He traveled for 60 minutes, or hour. The distance the second car traveled is D r t r. Since both cars traveled the same distance, we have: 6 r 6. r The second car drove at an average speed of. mph. 009 Duke University Talent Identification Program Page 4 of

15 . Give the linear equation, in slope-intercept form, of the line which 7, and is perpendicular to the line y. passes through ( ) y y + y The slope of the line is, so the slope of a line perpendicular is m. We can find the y-intercept using the slope-intercept formula: y m + b ( 7) + b b b The equation of the line is: y m + b y +. Solve the following systems of equations algebraically. a) 6 y 9y 009 Duke University Talent Identification Program Page of

16 6y 9y ( y ) ( y ) y 6 8y 4 0 No solution! b) 4 + y 7y 4 + y 7y ( + y ) 7 4 7y 8 + 7y 7y 0 If, then ( ) 4 + y 8 + y y Solution: (,) 009 Duke University Talent Identification Program Page 6 of

17 . Solve 6 0 by completing the square ( ) 6 ± 6 ± 6 4. Divide. Write the result as Quotient Re mainder +. Divisor ( 4 0 ) ( ) ( ) ( ) Are ( 6,4 ), (, ), and (, ) work and eplain your answer. 009 Duke University Talent Identification Program Page 7 of vertices of a right triangle Show all Label the points A, B, and C, respectively. We will find the distance between the three points and use the Pythagorean Theorem to determine if the points make a right triangle.

18 AB ( 6 ) ( 4 ) ( ) ( ) D + + BC ( ) ( ) ( ) ( ) D + + AC ( 6 ) ( 4 ( ) ) D + ( ) ( ) Since AC is the longest side, it would be the hypotenuse. We substitute the lengths of the sides into the Pythagorean Theorem and get: a + b c ( ) + ( ) ( 6 ) Therefore, the three points ARE the vertices of a right triangle. 6. Suppose your test grades for the semester in history are 9, 8, 9, and 8. The final eam for the semester counts as two of these test grades. What minimum score would you need to make on the final in order to have an average of 90 Let be the score for the final eam. To calculate the final average, we add all the scores together and divide by the total number of scores. Since the final eam counts as two tests, there are 6 total scores. These scores are 9, 8, 9, 8, and. The inequality we will use is: 009 Duke University Talent Identification Program Page 8 of

19 You would need a minimum score of 9. on your final to have a 90 average for the semester. 7. Student volunteers were selling tickets to the junior/senior prom. Junior tickets were $0 each, and senior tickets were $ each. The advisor asked how many juniors and how many seniors planned to attend the prom, but the student volunteers did not keep track. However, they knew that they had sold 66 tickets and that they had deposited $476. Write a system of equations to represent this information. Solve the system to determine the number of junior tickets and the number of senior tickets that were sold. Let j be the number of junior tickets and s be the number of senior tickets. 0j + s 476 j + s 66 0j + s 476 ( j s ) j + s 476 0j 0s 4980 If s 88, then 8 s 704 s Duke University Talent Identification Program Page 9 of

20 j + s 66 j j 78 There were 78 junior tickets sold and 88 senior tickets sold. 8. A boat travels mph in still water. If the boat travels 7 miles upstream in the same amount of time that it takes to travel 0 miles downstream, find the speed of the current. Let D U be the distance upstream, r U be the rate the boat travels upstream, and t U be the time in hours that the boat travels upstream. Similarly, D d, r d, and t d are the distance, rate and time, respectively, that the boat travels downstream. Let c be the speed of the current. When the boat travels against the current, its effective speed is equal to the speed of the boat minus the speed of the current. When the boat travels with the current, its effective speed is equal to the speed of the boat plus the speed of the current. D 7 r c U D 0 r + c d U d We know that D r t. Solving for t gives t D. r t U D r U U 7 c t d D r d d 0 + c Since it took the same amount of time to go upstream as it did to go downstream, we have that t U t d. Our equation is: 009 Duke University Talent Identification Program Page 0 of

21 t U t d 7 0 c + c We can solve by cross-multiplying: ( + c) ( c) c 840 0c 9c 6 c 8 The speed of the current is 8 mph. 009 Duke University Talent Identification Program Page of

MATH 110: FINAL EXAM REVIEW

MATH 110: FINAL EXAM REVIEW MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set