Polynomial graphs #3 - Questions and Answers

Question 1 Investigate the change in the graph of a sinusoidal function of the form 𝑦 = sin ⁡ 𝑥 y=sinx or 𝑦 = cos ⁡ 𝑥 y=cosx when multiplied by a polynomial function. Use a graphing calculator to sketch the graphs of 𝑦 = 𝑥 2 y=x 2 , 𝑦 = − 𝑥 2 y=−x 2 , and 𝑦 = 𝑥 2 sin ⁡ 𝑥 y=x 2 sinx on the same coordinate plane on the interval [ − 20 , 20 ] [−20,20]. Answer Here is the graph provided, showing how 𝑦 = 𝑥 2 sin ⁡ 𝑥 y=x 2 sinx interacts with sinusoidal and polynomial components. Question 2 Use a graphing calculator to examine the graphs of the following functions. Determine: the degree of the polynomial, the leading coefficient, the end behavior of the polynomial function, the maximum number of zeros, the maximum number of turning points (relative maxima and minima). a. 𝑓 ( 𝑥 ) = 1 2 𝑥 2 + 2 𝑥 f(x)= 2 1 ​ x 2 +2x b. 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) 2 ( 𝑥 − 1 ) 2 f(x)=x(x+1) 2 (x−1) 2 c. 𝑓 ( 𝑥 ) = 𝑥 3 − 𝑥 2 + 31 𝑥 + 40 f(x)=x 3 −x 2 +31x+40 Answer a. 𝑓 ( 𝑥 ) = 1 2 𝑥 2 + 2 𝑥 f(x)= 2 1 ​ x 2 +2x Degree: 2 2 (highest-degree monomial is 1 2 𝑥 2 2 1 ​ x 2 ) Leading coefficient: 1 2 2 1 ​ End behavior: 𝑦 → ∞ y→∞ as 𝑥 → ∞ x→∞, 𝑦 → ∞ y→∞ as 𝑥 → − ∞ x→−∞ (even-degree polynomial with positive leading coefficient). Maximum number of zeros: 2 2 (same as degree). Maximum number of turning points: 1 1 (one less than the degree). b. 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 + 1 ) 2 ( 𝑥 − 1 ) 2 f(x)=x(x+1) 2 (x−1) 2 Degree: 5 5 (product of 5 linear factors). Leading coefficient: 1 1. End behavior: 𝑦 → ∞ y→∞ as 𝑥 → ∞ x→∞, 𝑦 → − ∞ y→−∞ as 𝑥 → − ∞ x→−∞ (odd-degree polynomial with positive leading coefficient). Maximum number of zeros: 5 5 (same as degree). Maximum number of turning points: 4 4 (one less than degree). c. 𝑓 ( 𝑥 ) = 𝑥 3 − 𝑥 2 + 31 𝑥 + 40 f(x)=x 3 −x 2 +31x+40 Degree: 3 3 (degree of highest-degree monomial 𝑥 3 x 3 ). Leading coefficient: 1 1. End behavior: 𝑦 → ∞ y→∞ as 𝑥 → ∞ x→∞, 𝑦 → − ∞ y→−∞ as 𝑥 → − ∞ x→−∞ (odd-degree polynomial with positive leading coefficient). Maximum number of zeros: 3 3 (same as degree). Maximum number of turning points: 2 2 (one less than degree). Question 3 For the following exercise, for each polynomial: a. Find the degree. b. Find the zeros, if any. c. Find the 𝑦 y-intercept(s), if any. d. Use the leading coefficient to determine the graph's end behavior. e. Determine algebraically whether the polynomial is even, odd, or neither. 𝑓 ( 𝑥 ) = 𝑥 3 + 3 𝑥 2 − 𝑥 − 3 f(x)=x 3 +3x 2 −x−3 Answer a) Degree: 3. b) Zeros: 𝑥 = − 1 , 𝑥 = 1 x=−1,x=1. c) 𝑦 y-intercept: − 3 −3. d) Graph falls to the left and rises to the right. e) Function is neither even nor odd. Question 4 Write the equation of the polynomial given the following information: Degree 4. Root of multiplicity 2 at 𝑥 = 4 x=4, and roots of multiplicity 1 at 𝑥 = 1 x=1 and 𝑥 = − 2 x=−2. 𝑦 y-intercept at ( 0 , − 3 ) (0,−3). Answer Given data: 𝑥 x-intercept of multiplicity 2 at 𝑥 = 4 x=4. 𝑥 x-intercept of multiplicity 1 at 𝑥 = 1 x=1. 𝑥 x-intercept of multiplicity 1 at 𝑥 = − 2 x=−2. 𝑦 y-intercept at ( 0 , − 3 ) (0,−3). General equation: 𝑓 ( 𝑥 ) = 𝑎 ( 𝑥 + 2 ) ( 𝑥 − 1 ) ( 𝑥 − 4 ) 2 f(x)=a(x+2)(x−1)(x−4) 2 Substitute 𝑓 ( 0 ) = − 3 f(0)=−3: − 3 = 𝑎 ( 0 + 2 ) ( 0 − 1 ) ( 0 − 4 ) 2 −3=a(0+2)(0−1)(0−4) 2 − 3 = 𝑎 ( 2 ) ( − 1 ) ( 16 ) −3=a(2)(−1)(16) − 3 = − 32 𝑎 −3=−32a 𝑎 = − 3 − 32 = 3 32 a= −32 −3 ​ = 32 3 ​ Final equation: 𝑓 ( 𝑥 ) = 3 32 ( 𝑥 + 2 ) ( 𝑥 − 1 ) ( 𝑥 − 4 ) 2 f(x)= 32 3 ​ (x+2)(x−1)(x−4) 2 Question 5 Graph 𝑦 = 𝑒 𝑥 y=e x and 𝑦 = 1 + 𝑥 + 𝑥 2 2 + 𝑥 3 6 + 𝑥 4 24 y=1+x+ 2 x 2 ​ + 6 x 3 ​ + 24 x 4 ​ in the same viewing rectangle. Answer The exponential function graph 𝑦 = 𝑒 𝑥 y=e x : The polynomial function 𝑦 = 1 + 𝑥 + 𝑥 2 2 + 𝑥 3 6 + 𝑥 4 24 y=1+x+ 2 x 2 ​ + 6 x 3 ​ + 24 x 4 ​ : The two graphs are shown side-by-side in the same viewing rectangle (see diagrams for detailed comparison).