Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following statements is true. Function f(x) Function g(x) g(x) = (x 8) 3 A. The functions have the same vertex. B. The minimum value of f(x) is the same as the minimum value of g(x). C. The functions have the same axis of symmetry. D. The minimum value of f(x) is less than the minimum value of g(x). ) Simplify: (i 5 + 3) A. 4 + 6i 5 C. 6i 5 B. 4 + 6i 10 D. 14 GOAL: Simplify and multiply rational expressions Simplify the following rational expressions. 1. 3x 1 3. 4 18x y z 3 9xy z 3. 4y 8y y 4
4x 0 4. x 11x 30 5. x 4 6 3(x 5) x(x 4) 6. x 10 9x x 3x 10 6x 1x 7. 5k 10k 6k 6 k 15k 30 8. 3y 1 4y 400 1y 10 y 8y 0 x 4 9. x x 1 3 x 1 x 1x 18x 4 3 10. x 3x 9 3 x 7
Review Questions Mission Divide Rational Expressions 1. The equation x 3 3x + 4x 1 = 0 is graphed below. Use the graph to help solve the equation and find all the roots of the function. A. x = 3,, B. x = 1, 1, 3 C. x = 3, i, i D. x = 1, 3 i 7, 3 + i 7. Which function is represented by the graph? A. f(x) = 1 (x 3)(x 6) 3 B. f(x) = (x + 3)(x + 6) C. f(x) = (x 3)(x 6) D. f(x) = 3(x 3)(x 6) 3. Simplify +i 3i 4. Use a graphing calculator and then state the end behavior of f(x) = 4 x GOAL: Simplify and divide rational expressions. Simplify the following rational expressions. a a 1. 3a 6 a 4. 7x 1x x 3x x 4x 1
5y 0 4. 0y 3. y 16 k m 6k 6m k km 3m 3 5. 3 8x 8x x 7 x 6x 7 6. k 5 k 11k 5 9k 18k k 5k
Review Questions Mission 3 Add and Subtract Rational Expressions 1. Factor the following using imaginary numbers: 16x + 49 A. (4x 7) C. (4x + 7i)(4x 7i) B. ( 4x + 7)( 4x 7) D. (4x + 7i). Simplify: x 9x+14 x 6x+5 x 8x+7 x 7x+10 A. B. (x 7) (x 5) C. (x 5)(x 7) (x 1) (x ) (x 1) D. (x 7) (x 1) GOAL: Add and Subtract rational expressions Simplify the following rational expressions. 1. 3a 1 a 5. 4 4 y 1 y 1 1y 3. x 3x 10 x 5 5 x 4. 5 3 3x 4
4 6 5. 6. 4m m x 1 x 1 x 7. m 4 m m 4 x 1 4x x 4x 8. x 1 6 x 6x 9 x 9 9.
Mission 4 Complex Fractions Review Questions 1. Write 7i(1+i)+5 6i as a complex number in standard form. A. B. 3i 3 9 + 7i 6i C. D. 7 6 + 3i 7 + 5 6 i. Simplify: 1 1 x + x x 1 A. 1 C. x + 1 1 x B. x + 1 x 1 D. x + 1 (x 1) 3. Solve over the set of complex numbers: (x 8) = -16 GOAL: Simplify complex fractions Complex fractions - Simplify the following rational expressions. 1. 8 5x. x 1 4 15 15 x x1 1 3x x 1
3. 6y y 4 4 8 3y 4. 5 7 x x 3 1 x x 5. Perform the indicated operation: x 3 4 x+1 +x 3 A. 3x + 3 (x + 4) C. 3x 6x 9 8x B. x 3 x 15x + 36 6(x + 1) D. 3x 6x 9 ( 4 + x)
Mission 5 Solve Rational Equations Review Questions 1. Perform the indicated operation: x+4 x+8 + x 1 x 3 5x 6 x +5x 4 A. x + 3x + 41 (x + 8) (x 3) C. x + 3x 14 (x + 8)(x 3) B. 10x x 1 (x + 8)(x 3) D. 3x + 9 (x + 8)(x 3). What is the end behavior for the function, f(x) = (x 11 5x 3)(9x 5 + 6x 3 )? A. as x, f(x) and as x +, f(x) + B. as x, f(x) + and as x +, f(x) + C. as x, f(x) and as x +, f(x) D. as x, f(x) + and as x +, f(x) Goal: Solve rational equations and report excluded values Solve the following rational equation by cross multiplying. Then state any excluded values. 6 1 1. x x 1 State the excluded values of each rational function. Write the domain.. f(x) = (x+)(x 3) 3. g(x) = 3x 5x(x+1)(x 4)(x+5)
3 STEPS TO SOLVE EQUATIONS CONTAINING MULTIPLE RATIONAL EXPRESSIONS 1) List the excluded values for each rational part of the equation. In other words, set each denominator =. ) Multiply by the least common denominator (LCD) for the entire equation (both sides). 3) Check to see if any of your solutions make the denominators of your problem 0. This means these values are excluded values and will produce an undefined value. Solve the following rational equations. List excluded values first. a 3 a 1 5x 1 4. 5. 5 6 3 x 1 x Excluded Values? Excluded Values? 6. y 3 1 y 3 y 3 3x 5 8 7. x 3 x x 1 Excluded Values? Excluded Values?
m 3 m 8. 1 m 1 m 1 9. 6x 3x 4 x 16 x 4 x 4 Excluded Values? Excluded Values? Practice on your own 10. Solve: = 1 x 4 x 4 A. x = C. x = B. x = 0 D. no solution 11. Solve: x 1 x+1 + x+7 x 1 = 4 x 1 A. x = 1, C. x = B. x = 1, 1 D. no solution
Mission 6 Graph the Parent Rational Function & Its Transformations Review Questions 1. What is the remainder in the division (6x 3 x + 5x 9) (x 5)? A. 11 C. 91 B. 91 D. 109. Solve: 10y 3 4y y = 5y 3 + 3y A. y = 3, y = 0, y = 10 C. y = 0, y = 1 ± 41 10 B. y = 1 5, y = 0, y = 3 D. y = 0 GOAL: Graph rational functions. Translate rational functions. Find asymptotes of a rational function. The Parent Rational Function f(x) = 1 x x y Vertical Asymptote: Horizontal Asymptote: Sketch the graphs of these functions on your graphing calculator and describe the transformation. f(x) = 1 x f(x) = 3 x f(x) = 1 x + 3 f(x) = 1 x+ f(x) = 1 x 3
(h, k) form of the parent rational function f(x) = 1 x f(x) = a x h + k Vertical Asymptote: Horizontal Asymptote: When a is positive When a is negative Finding asymptotes In order to find the vertical asymptote of any rational function, set the = and solve. In order to find the horizontal asymptote of a rational function in graphing form then consider that the graph will move up or down units and the result will always be y =. State the asymptotes, the domain, the range and the end behavior of each function. 1) f(x) = 3 x + 5 ) f(x) = x+7-11 VA VA VA HA HA HA 1 3) f(x) = + 6 x 4 Domain Domain Domain Range Range Range End End End
Sketch the graph of each rational function and list the asymptotes. 4) f(x) = x+1 3 1 5) f(x) = + 1 x (A table isn t always needed.) x y VA HA VA HA
Review Questions Mission 7 - Rational Functions not in Graphing Form 1 Which of the following functions is modeled by the graph below? A. f(x) = x+3 + 4 B. f(x) = x + 3 + 4 C. f(x) = x 3 + 4 D. f(x) = x 3 + 4. What is the maximum number of horizontal asymptotes a rational function can have? GOAL: Graph rational functions in any form. Find asymptotes and holes. Rewrite rational functions in graphing form. STEPS TO GRAPH RATIONAL FUNCTIONS Find the vertical asymptotes. Find the horizontal asymptotes. Find any holes. Find any x-intercepts. Create a table by picking x-values near the vertical asymptotes. Use symmetry to get other points. Use your graphing calculator to check.
How to find asymptotes, holes and x-intercepts Given the function p(x) f (x) = q(x) poly with degree n poly with degree m Vertical Asymptote: To find the V.A. set equal to and solve. Be sure that you have already cancelled any repeated factors before doing this. Horizontal Asymptote: Holes: To find the H. A. there are three cases o Case 1: n < m o Case : n = m o Case 3: n > m The H.A. is. The H.A. is, where and are leading coefficients of the numerator and denominator H.A does not. Holes are present when factors are. Set the common factor equal to zero and solve for x. x-intercepts: To find the x-intercepts set equal to and solve. Find the asymptotes for each rational function. 1. y = 5 x. y = x+3 x 8 3. y = 4 x x+1 4. y = x + 4 5. y = 4 x x +1 VA VA VA VA VA HA HA HA HA HA Find the x values for which each rational function has a hole. Also list any asymptotes. x 9 3x 6x 5. y 6. y x 7x1 ( x 16)( x)
Graph each rational function. x 4 7. y x 1 8. y = x x(x+) x y Vertical Asymptote: Horizontal Asymptote: Vertical Asymptote: Horizontal Asymptote: Hole @ x = x-intercept(s): End Behavior: x-intercept(s): End Behavior Find the graphing form of each of the above by using synthetic division. Notice that the graphing form is displaying the rational function in terms of how it relates to the parent function y = 1 x.
Mission 8 Graph Rational Functions in Any Form Review Questions Which is a graph of f(x) = 4x+4 with any asymptotes indicated by dashed 1. x+ lines? A. C. B. D.. Which statement describes the end behavior of the function f(x) = 5x+4 x 3? A. as x, f(x) + 3 and as x +, f(x) 5 B. as x, f(x) and as x +, f(x) + 3 C. as x, f(x) 5 and as x +, f(x) 5 D. as x, f(x) and as x +, f(x) 5 3. Which of the following is the graphing form of f(x) = 4x 14 x 6? A. f(x) = 6 4 + 10 C. f(x) = x 3 x 6 + 4 B. f(x) = 4 10 + 10 D. f(x) = x 3 x 6 + 4 GOAL: Graph rational functions in any form.
Graph each rational function. State the domain and range. Check for any holes. 1. f(x) = x+1 x 4. f(x) = 1 x+ + 1 x y x y Domain: Domain: Range: Range: 3. f(x) = x 1 x 4x+3 4. y x x 4 4x4 x y x y Domain: Range: Hole(s): Domain: Range: Hole(s):
5. Create a new rational function g(x) that moves the given function f(x) up 4 and left 5 units. f(x) = 1 x+3 9 g(x) = 6. Create a new rational function g(x) that moves the given function down and right 6 units. f(x) = x+1 x+5 g(x) = Graphing Form
Review Questions Mission 9 Rational Functions & (Slant Asymptotes Optional) 1) List the asymptotes of y = x+9 x 3 ) State the end behavior of the graph of y = x + 3 3) Identify any holes, asymptotes, and intercepts of f(x) = x x 6 x +7x+10 A. Horizontal Asymptote: y =, 3 Vertical Asymptote: x = 5, Hole: none x-intercept: (10, 0) y-intercept: (0, 6) C. Horizontal Asymptote: y = 1 Vertical Asymptote: x = 5 Hole at x = x-intercept: (3, 0) y-intercept: (0, 3 5 ) B. Horizontal Asymptote: none Vertical Asymptote: x = 5 Hole at x = x-intercept: (3, 0) y-intercept: (0, 5) D. Horizontal Asymptote: y = 5 Vertical Asymptote: x = 1 Hole: none x-intercept: (, 0), ( 5, 0) y-intercept: (0, ), (0, 3) GOAL: Graph all rational functions including those with slant asymptotes. 5 1. y x 1. f(x) = x 4 x y x y Vertical Asymptote(s): Horizontal Asymptote: Vertical Asymptote(s): Horizontal Asymptote:
3. f(x) = 3x + 1 4. f(x) = x x+1 x x x 1 x y x y Vertical Asymptote(s): Horizontal Asymptote: Hole(s): Vertical Asymptote(s): Horizontal Asymptote: Hole(s):
Optional - Graphs with Slant Asymptotes A slant asymptote occurs when the degree of the numerator is than that of the denominator. In order to find the equation of the slant, do synthetic division when the denominator is. Otherwise, you will need to perform long division. List the vertical, horizontal, and slant asymptotes of each. 5. x y x 1 6. y = x 1 x 3 x y x y Vertical Asymptotes: Horizontal Asymptotes: Slant: x-intercepts: Vertical Asymptotes: Horizontal Asymptotes: Slant: x-intercepts: