UMUC MATH-107 Final Exam Information
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1 UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from the book and work all 30 problems on the attached practice exam. The detailed instructor solutions are at the end of this package. The actual final exam is VERY close to this practice exam. Section Material R. Reducing fractions with exponents (powers). Circle equations, Distance formula, Midpoint formula. Domain and Range of functions.3 Slope (given two points or an equation).4 Equation of lines (from two points, or a point and a slope).5 Decreasing and Increasing intervals.6 Composition functions. Linear equation word problems. Complex numbers.3 Quadratic functions.4 Parabola equations (finding vertex by completing the square).5 Solving equations with sqaure roots and fractions.6 Solving inequality equations 3. End Shape of functions 3.3 Long division of polynomials 3.4 Zeros of a function 3.5 Asymptotes (vertical and horizontal), Domain of fractions 3.6 Solving inequality equations (using interval tests) 4. Inverse functions 4. Graphs of exponents and exponent word problems 4.4 Log Properties 4.5 Solving Exponent and Logarithm equations
2 MATH 07 PRACTICE EXAM Closed Book, Calculator Permitted. NOTE: Equation Sheet and Solutions at the End MULTIPLE CHOICE. Find the center and radius of the circle with equation ( x 6) + ( y + 5) = 4 A. ( 6, 5); 4 B. ( 6, 5); C. (6, 5); 4 D. (6, 5);. The points (, ) and (4, 3) are endpoints of the diameter of a circle. Find the length of the radius of the circle. A. B. C. 0 D Find the midpoint of the segment with endpoints (6, ) and ( 4, 7). A. ( 5, 3) B. (, 4) C. 7 3, D. (5, 3) 4. Simplify: 5a 8 A. 0a b B. 0a b 4 C. 5a 0 b D. 5a 0 b 5 a b 7 ( a b )
3 5. Which of the following equations does the graph represent? A. x + y = B. x + y = C. x y = D. x y = 6. (7 pts) Suppose it costs $8.00 per day plus $0.5 per mile driven to rent a car. If the total rental charge for one day is $00.50, how many miles were driven? A. 330 B. 40 C. 474 D Find the slope of the line through the points (5, 6) and (, 6). A. 7 B. 0 C. 4 D. Undefined 8. Write a slope-intercept equation for a line passing through the point (4, ) and parallel to the line y = 3x + 5. A. y = 5x 9 B. y = 3x + C. y = 3x D. y = 3x +
4 For #9 and #0, consider the following graph What are the domain and the range? A. Domain [ 4, ] and Range [, ] B. Domain [ 3, ] and Range [, ] C. Domain [ 3, 0] and Range [0, ] D. Domain [, ] and Range [ 4, ] 0. On what interval is the function decreasing? A. ( 4, ) B. ( 3, 0) C. (, 0) D. (, ). Which of the following functions has a graph which is symmetric with respect to y-axis? A. f(x) = 6x B. f(x) = x C. f(x) = x + 4 x D. f ( x) = x 3. Given f (x) = x 3x + 5 and g(x) = x, find the composite function ( f o g)( x) and simplify. A. x 7x + 5 B. x 3x C. x 3x + 9 D. x 3 5x + x 0
5 3. Simplify (8 + 3i) and write the answer in the form a + bi, where a and b are real numbers. A. 55 B. 73 C i D i 4. Find the exact solutions: x x 5 = 0. A., 5/ B i, + i C. 4 4, D. None of the above 5. Find the vertex for the graph of the function f (x) = x + 6x + 7. A. ( 3, ) B. ( 3, ) C. ( 3, 6) D. ( 3, 6) 6. Solve: x = x A., B. C., D. 7. Given f ( x) = x 5 and g ( x) = x 7, find the domain of the quotient function f / g. A. (, 5) ( 5, ) B. (, 7] C. (, 7) ( 7, ) D. ( 7, )
6 x 8. Solve x + = x x + A. B. C. 3 D. No solution 9. Which of the following describes the end behavior of the graph of the function f(x) = 6x 3 + x + 5x? A. Upward to the left and upward to the right vvvv B. Downward to the left and downward to the right vvvv C. Downward to the left and upward to the right vvvv D. vvvv Upward to the left and downward to the right 0. Find the quotient and remainder when 4x 3 3x 9x 5 is divided by x. A. 4x x + 3, Remainder 67 B. 4x 5x + 9, Remainder 33 C. 4x + 5x +, Remainder 3 D. 4x + x + 3, Remainder 67
7 . Find all of the zeros of the polynomial x 3 + 3x 8x + 3. A.,, 3 B.,, 3 3 C. 3, D. 3,,. Find all vertical asymptotes for the rational function A. x = x 9 f ( x) =. x B. x = 9 C. x = 3 and x = 3 D. x = and x = 3. Solve, and write the answer in interval notation: x(x 4) 0 A. [ 0, ) B. [ 0, 4] C. [ 4, ) D. (, 0] [ 4, ) 4. Solve, and write the answer in set notation: 7 x < 9. A. {x < x < 8} B. {x x < } C. {x x > 8} D. {x x < or x > 8} 5. Given the function f ( x) = 3x + 5, find a formula for the inverse function. A. f ( x) B. f ( x) C. f ( x) x 3 = 5 x 5 = 3 = 3x + 5 D. f ( x) = 3x 5
8 6. Determine which is the graph of y = + e x. A. B. C. D.
9 .t 7. The value of a stock is given by the function V ( t) ( e ) = where V is the value of the stock after time t, in months. Find the value of the stock after months. A B C D Solve: x = log 3 7 A. 9 B. 3 C. 3 D. 9 9.Express as a single logarithm: 3 log + ½ log p log q. 8 p A. log q B. 8 + log q p C. log ( 8 + p q) D. log ( 3pq) 3x Solve 4 = 6. A. B. C D. 3
10 MATH 07 FORMULAS Properties of exponents a p a q = a p + q a p / a q = a p q (a p ) q = a p q (ab) p = a p b p Factoring (a + b) = a + ab + b (a b) = a ab + b a b = (a + b)(a b) a 3 + b 3 = (a + b)(a ab + b ) a 3 b 3 = (a b)(a + ab + b ) x for x 0 Absolute value: x = x for x < 0 Distance The distance between a and b on the number line is a b. The distance between points (x, y ) and (x, y ) is given by ( ) ( ) x +. x y y Midpoint Formula: If the endpoints of a line segment are (x, y ) and (x, y ) then the coordinates of the x + x y + y midpoint are, Equation of a Circle: The equation of a circle with center (h, k) and radius r, in standard form, is ( x h) + ( y k) = r Slope The slope of a line containing points (x, y ) and (x, y ) where x x, is given by y y m = x x. Slope-Intercept Equation: y = mx + b, where m is the slope and (0, b) is the y-intercept. Point-Slope Equation: y y = m(x x ), where m is the slope and (x, y ) is a point on the line. Composition of functions: ( f g)( x) = f ( g(x) ) o where x is in the domain of g and g(x) is in the domain of f. Symmetry of functions Even function Graph symmetric with respect to y-axis f(x) = f( x) for all x in the domain Odd function Graph symmetric with respect to origin f(x) = f( x) for all x in the domain Transformation of function y = f(x) Vertical translation: Graph of y = f(x) + b is the graph of y = f(x) shifted upward b units Graph of y = f(x) b is the graph of y = f(x) shifted downward b units Horizontal translation Graph of y = f(x d) is the graph of y = f(x) shifted rightward d units Graph of y = f(x + d) is the graph of y = f(x) shifted leftward d units Reflection Graph of y = f(x) is the reflection of the graph of y = f(x) across the x-axis Graph of y = f( x) is the reflection of the graph of y = f(x) across the y-axis Vertical Stretching or Shrinking: Graph of y = af(x) is a vertical stretching/shrinking of the graph of y = f(x) Horizontal Stretching or Shrinking Graph of y = f(cx) is a horizontal stretching/shrinking of the graph of y = f(x)
11 Complex number: a + bi, where a and b are real numbers and i = Quadratic Formula b ± x = b 4ac a Quadratic Functions f(x) = ax + bx + c b f(x) = a(x h) b + k h =, k = f = f( h) a a Absolute Value Inequalities For a > 0, x < a a < x < a For a > 0, x > a x < a or x > a Polynomials Polynomial Function: P(x) = a n x n + a n- x n- + a n- x n- + + a x + a 0 Intermediate Value Theorem: For any polynomial P(x) with real coefficients, if P(a) and P(b) have opposite signs, then the function P must a real zero between a and b. Remainder Theorem: Given polynomial f(x) and value c, when dividing f(x) by x c, the remainder is f(c). That is, f(x) = (x c) Q(x) + R, where R = f(c). Factor Theorem: For a polynomial f(x), if f(c) = 0, then x c is a factor of f(x). Fundamental Theorem of Algebra: Every polynomial function of degree n, with n, has at least one zero in the system of complex numbers. Polynomial with real coefficients For a nonzero, a + bi is a zero a bi is a zero. Polynomial with rational coefficients For a and c rational and b not a perfect square, a + c b is a zero a c b is a zero. Polynomial with integer coefficients Rational Zeros Theorem: If the coefficients are all integers, then for any rational zero (which is a zero of the form p/q) the numerator p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n. Descartes Rule of Signs The number of positive real zeros of P(x) is either the same as the number of variations of sign in P(x) or less than the number of variations of sign in P(x) by a positive even integer. The number of negative real zeros of P(x) is either the same as the number of variations of sign in P( x) or less than the number of variations of sign in P( x) by a positive even integer. A zero of multiplicity m must be counted m times. Rational Function: A quotient of two polynomials, q(x) is not the zero polynomial. p( x) f ( x) =, where p(x) and q(x) are polynomials and q( x) Direct Variation y = kx, where k is a positive constant Inverse Variation k y =, where k is a positive constant x Exponential Function with base a: f(x) = a x, where x is a real number, a > 0 and a. Logarithms y = log a x x = a y log a (MN) = log a M + log a N log M = log N M log log a (M p ) = p log a M log a = 0 log a a = log a a x = x a a a N
12 SOLUTIONS. D. C 3. B 4. D 5. B 6. A 7. B 8. C 9. A 0. A. B. A 3. C 4. C 5. B 6. D 7. D 8. A 9. C 0. C. D. D 3. D 4. A 5. B 6. B 7. C 8. B 9. A 30. C
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