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MHF4U Characteristics of Polynomials Quiz 1 NAME: ________________ Please do not use graphing calculators for this assessment. 1. Consider a polynomial relation of the form y=−x5 + a x 3 +bx +c , where a , … , c ∈ R. Fill in the requested information in the table below. End behaviour Domain Type of range Number of Number of (circle one): zeros turning points As x →−∞ , Describe (list all (list all poly { y ∈ R} y→ possibilities): possibilities): K/A State { y ∈ R∨ y ≥ k } Features As x → ∞ , { y ∈ R∨ y ≤ k } y→ 2. a) Check off all relations that could represent polynomial functions. Note: each graph shows all the turning points for its function. 3 y−2 x =5 2 □ □ □ □ □ □ □ b) Determine whether the table of values below represents a polynomial function. If so, state its degree; if not, explain why not. Show supporting work. Compare poly and non-poly fn’s y=3 x 3 +2 x+ x−1 x 2+ y 2=1 y=2x +4 x -2 -1 y -16 -1 0 0 1 1 2 16 K/A Classify 3. Match the graphs to their equations. Label each graph by writing the capital letter of the matching function (A, B, C, or D). A(x )=x (x +1)2 ( x−1)3 B( x)=x 2(x +1)3 (x−1) T C (x)=x 3 (x +1)( x−1)2 Determine Eqn from Graph D( x )=x ( x−1)2(x +1)3 Match 4. a) Sketch a polynomial function y=f ( x )that satisfies all of the following conditions: There is a zero of order 2 at x=2 There is a zero of order 1 at x=1 There is a zero of order 3 at x=−3 As x → ± ∞ , y →−∞ b) Determine an equation in factored form of the function from part (a) if it passes through the point (0, 54). A Determine poly fn from conditions Sketch K C Equation Neatness and Organization 5. For each function given below, determine whether the function has even symmetry, odd symmetry, or neither. Show your reasoning. a) 9 8 7 6 5 4 3 2 1 -9 -8 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7 -8 -9 b) f (x)=x2 (x 3−1)( x 3 +1) x 1 2 3 4 5 6 7 8 9 Determine Symmetry K Understanding concepts, procedures C Clarity and Conventions MHF4U Characteristics of Polynomials Quiz 1 NAME: ________________ Please do not use graphing calculators for this assessment. 1. Consider a polynomial relation of the form y=−x 4 +ax 3 +b x 2+ cx+ d , where a , … , d ∈ R. Fill in the requested information in the table below. End behaviour Domain Type of range Number of Number of (circle one): zeros turning points As x →−∞ , (list all (list all { y ∈ R} y→ possibilities): possibilities): { y ∈ R∨ y ≥ k } As x → ∞ , { y ∈ R∨ y ≤ k } y→ Describe poly K/A State Features 2. a) Circle all relations that could represent polynomial functions. Note: each graph shows all the turning points for its function. □ □ □ □ □ □ □ x= y 2 +1 y=3 x 3 +1 x 1−1 x−1 3 y−2 x =5 2 y=3 x +2 x +1 x b) Determine whether the table of values below represents a polynomial function. If so, state its degree; if not, explain why not. Show supporting work. Compare poly and non-poly fn’s x -3 -2 -1 0 1 y 34 10 0 -2 -2 K/A Classify 3. Match the graphs to their equations. Label each graph by writing the capital letter of the matching function (A, B, C, or D). A(x )=x (x +1)2 (x−1)3 B( x)=x 2(x +1)3 (x−1) T C (x)=x 3 ( x +1)(x−1)2 Determine Eqn from Graph D( x )=x (x−1)2( x +1)3 Match 4. a) Sketch a polynomial function y=f ( x )that satisfies all of the following conditions: There is a zero of order 2 at x=−2 There is a zero of order 1 at x=1 There is a zero of order 3 at x=3 As x → ± ∞ , y →−∞ b) Determine an equation in factored form of the function from part (a) if it passes through the point (0,−54). A Determine poly fn from conditions Sketch K C Equation Neatness and Organization 5. For each function given below, determine whether the function has even symmetry, odd symmetry, or neither. Show your reasoning. a) 9 8 7 6 5 4 3 2 1 -9 -8 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7 -8 -9 b) f (x)=x2 (x 3−1)( x 3 +1) x 1 2 3 4 5 6 7 8 9 Determine Symmetry K Understanding concepts, procedures C Clarity and Conventions MHF4U Characteristics of Polynomials Quiz 2 NAME: ________________ Please do not use graphing calculators for this assessment. 1. Consider a polynomial relation of the form y=an xn + an−1 x n−1 +…+a 2 x 2+ a1 x+ a0, where a i ∈ R . Answer each question below: Suppose as Can the domain be Can the range be How many xHow many x → ± ∞ , y →∓ ∞. {x ∈ R∨x ≥ 0 }? If yes, turning points { y ∈ R∨ y ≥ 50 } ? If yes, intercepts are Under which possible if n is are possible if state the conditions; if state the conditions; if conditions would odd? n is even? not, explain why not. not, explain why not. this be true? Describe poly K/A State Features 2 . State 2 features of the following graph that indicate that it is not a polynomial relation. Compare poly and non-poly fn’s Check off the equations that represent polynomial relations. □ □ □ □ □ K/A y=2x + x1 +10 Determine whether the table of values below represents a polynomial relation of degree 5. Show your reasoning. x 4 6 8 10 12 14 16 y 7 7 6 2 -3 -6 0 √ y=x 2 +1 y+ x =1 2 y+ 2 =1 x xy=1 Classify 3. a) Sketch a polynomial function y=f ( x )of degree 4 that satisfies all of the following conditions: There is a zero of order 2 at x=−2 f ( 1 ± √2 ) =0 f ( 1 ) =3 K/A Compare poly and non-poly fn’s Classify b) Determine an equation in factored form of the function from part (a). A Determine poly fn from conditions Sketch K C Equation Neatness and Organization 4. For each function given below, determine whether the function has even symmetry, odd symmetry, or neither. Show your reasoning. a) b) f (x)=x3 (x 2−x )11 Determine Symmetry K Understanding concepts, procedures C Clarity and Conventions MHF4U Characteristics of Polynomials Quiz 2 NAME: ________________ Please do not use graphing calculators for this assessment. 1. Consider a polynomial relation of the form y=an xn + an−1 x n−1 +…+a 2 x 2+ a1 x+ a0, where a i ∈ R . Answer each question below: Suppose as Can the domain be Can the range be How many xHow many x → ± ∞ , y →−∞. {x ∈ R∨x ≥ 0 }? If yes, turning points { y ∈ R } ? If yes, state the intercepts are Under which possible if n is are possible if state the conditions; if conditions; if not, conditions would even? n is odd? not, explain why not. explain why not. this be true? Describe poly K/A State Features 2 . State 2 features of the following graph that indicate that it is not a polynomial relation. Compare poly and non-poly fn’s Check off the equations that represent polynomial relations. □ □ □ □ □ K/A y=2x + x1 +10 Determine whether the table of values below represents a polynomial relation of degree 5. Show your reasoning. x 4 6 8 10 12 14 16 y 7 7 6 2 -3 -6 0 √ y=x 2 +1 y+ x =1 2 y+ 2 =1 x xy=1 Classify 3. a) Sketch a polynomial function y=f ( x )of degree 4 that satisfies all of the following conditions: There is a zero of order 2 at x=−2 f ( 1 ± √ 3 )=0 f ( 1 ) =9 K/A Compare poly and non-poly fn’s Classify b) Determine an equation in factored form of the function from part (a). A Determine poly fn from conditions Sketch K C Equation Neatness and Organization 4. For each function given below, determine whether the function has even symmetry, odd symmetry, or neither. Show your reasoning. a) b) f (x)=x3 (x 2−x )10 Determine Symmetry K Understanding concepts, procedures C Clarity and Conventions MHF4U0 Polynomial Algebra Quiz 1 NAME:_________ 1. Determine the remainder when x 160 −2 x 3 +5 is divided by x +1. K Understand remainder theorem Calculations 2. Use the most efficient method to factor each of the following polynomials. Clearly show your procedures. a) 4 x 4 +12 x 2 +9 b) 40 x 4 + 96 x3 +5 x +12 3. Determine the number of rational and irrational zeros for the function f ( x )=4 x 4 + x 2−3 x +1. 4. Suppose two rectangular prisms with the dimensions shown below have equal volume. Determine this volume. Make sure to check that your solution makes sense in this context. Show your strategy and calculations. x K Work with polynomial equations 5. Factoring strategies A Solve polynomial equations C Organization Interpret results and conventions Determine if x=−2 satisfies the inequality 2 x−3 ≥ 4 x+ 6. Show your strategy. C (x) 6. 7. T Use the graph on the right to determine the values of x for which P ( x )<C ( x ). Show your solution set along a number line and write the solution using an inequality or interval notation. Demonstrate a strategy for solving each inequality below algebraically. a) 2 x+5> 7 x −3 b) (2−3 x ) ( x−1 )2 (−x 2−2 x +2)≤ 0 Work with polynomial inequalities K Check inequality solution A Solve inequalities graphically K Solve inequalities algebraically C Organization and conventions P( x ) MHF4U0 Polynomial Algebra Quiz 1 NAME:_________ 1. Determine the remainder when x 161−2 x 3 +5 is divided by x +1. K Understand remainder theorem Calculations 2. Use the most efficient method to factor each of the following polynomials. Clearly show your procedures. a) 4 x 4−12 x 2+ 9 b) 40 x 4 −96 x 3+ 5 x −12 3. Determine the number of rational and irrational zeros for the function f ( x )=4 x 4 + x 2 +3 x+1. 4. Suppose two rectangular prisms with the dimensions shown below have equal volume. Determine this volume. Make sure to check that your solution makes sense in this context. Show your strategy and calculations. x K Work with polynomial equations 5. Factoring strategies A Solve polynomial equations C Organization Interpret results and conventions Determine if x=−2 satisfies the inequality 2 x−3 ≥ 4 x+ 6. Show your strategy. C (x) 6. 7. T Use the graph on the right to determine the values of x for which P ( x ) ≥ C( x ). Show your solution set along a number line and write the solution using an inequality or interval notation. Demonstrate a strategy for solving each inequality below algebraically. a) 2 x+5> 7 x −3 b) (2−3 x ) ( x−1 )2 (−x 2−2 x +2)≥ 0 Work with polynomial inequalities K Check inequality solution A Solve inequalities graphically K Solve inequalities algebraically C Organization and conventions P( x ) MHF4U0 Polynomial Algebra Quiz 2 NAME:_________ 1. Determine the value of k if the remainder when x 4 +k x 3 +5 is divided by 2 x+1 is equal to 95 . 16 K Understand remainder theorem Calculations 2. Factor each polynomial below fully without repeating any strategies. Clearly show your procedures. a) 64 x 6−16 x 3 +1 b) 9 x 4 +6 x 3 +10 x2 +6 x +1 3. The equation h ( d )=0.1 d(d−20)2, 0 ≤ d ≤ 20 models the height of a roller coaster, h, with respect to the horizontal distance from the starting point, d (all measurements are in meters). The coaster operator wishes to place a camera on the coaster track 100 m above ground to snap pictures of the riders on their way up on the first slope. Determine the horizontal distance from the starting point at which the camera should be placed. Make sure to check that your solution makes sense in this context. Show your strategy and calculations. K Work with polynomial equations Factoring strategies A Solve polynomial equations T C Organization Interpret results and conventions MHF4U0 Polynomial Algebra Quiz 2 NAME:_________ 1. Determine the value of k if the remainder when x 4 +k x 3 +5 is divided by 3 x+ 1 is equal to 379 . 81 K Understand remainder theorem Calculations 2. Factor each polynomial below fully without repeating any strategies. Clearly show your procedures. a) 9 x 4 −6 x3 +28 x 2−18 x+ 3 b) x 6 +54 x 3+ 729 3. The equation h ( d )=0.05 d (d−3 0)2, 0 ≤ d ≤ 3 8 models the height of a roller coaster, h, with respect to the horizontal distance from the starting point, d (all measurements are in meters). The coaster operator wishes to place a camera on the coaster track 100 m above ground to snap pictures of the riders on their way up on the second slope. Determine the horizontal distance from the starting point at which the camera should be placed. Make sure to check that your solution makes sense in this context. Show your strategy and calculations. K Work with polynomial equations Factoring strategies A Solve polynomial equations T C Organization Interpret results and conventions