Question 1 How do you divide 6 − 4 𝑖 9 − 4 𝑖 2 ? 9−4i 2 6−4i ? Answer 1 Step 1 Always get rid of the complex numbers on the bottom. This can be done by multiplying the denominator and the numerator by the complex conjugate of the denominator. Allow me to demonstrate: 6 − 4 𝑖 9 − 4 𝑖 2 = 6 − 4 𝑖 9 − 4 𝑖 × 9 + 4 𝑖 9 + 4 𝑖 9−4i 2 6−4i = 9−4i 6−4i × 9+4i 9+4i Step 2 Doing so gets rid of the imaginary part of the denominator. 6 − 4 𝑖 9 − 4 𝑖 × 9 + 4 𝑖 9 + 4 𝑖 = ( 6 − 4 𝑖 ) ( 9 + 4 𝑖 ) ( 9 ) 2 − ( 4 𝑖 ) 2 9−4i 6−4i × 9+4i 9+4i = (9) 2 −(4i) 2 (6−4i)(9+4i) = ( 6 ⋅ 9 ) + ( 6 ⋅ 4 𝑖 ) − ( 4 𝑖 ⋅ 9 ) − ( 4 𝑖 ⋅ 4 𝑖 ) 81 − 16 ( − 1 ) = 54 + 24 𝑖 − 36 𝑖 + 16 97 = 70 − 12 𝑖 97 = 81−16(−1) (6⋅9)+(6⋅4i)−(4i⋅9)−(4i⋅4i) = 97 54+24i−36i+16 = 97 70−12i = 70 97 − 12 97 𝑖 = 97 70 − 97 12 i Question 2 How do you simplify 2 𝑖 5 ( 𝑖 − 7 𝑖 15 ) 2i 5 (i−7i 15 )? Answer 1 Step 1 2 𝑖 5 × ( 𝑖 − 7 𝑖 15 ) = 2 𝑖 × ( 𝑖 − 7 𝑖 3 ) = 2 𝑖 × ( 𝑖 − 7 𝑖 × 𝑖 2 ) = 2 𝑖 × ( 𝑖 + 7 𝑖 ) = 2 𝑖 × 8 𝑖 = 16 𝑖 2 = − 16 2i 5 ×(i−7i 15 )=2i×(i−7i 3 )=2i×(i−7i×i 2 )=2i×(i+7i)=2i×8i=16i 2 =−16 In the calculations I used the following properties of the imaginary unit: 𝑖 2 = − 1 i 2 =−1 𝑖 4 = 1 i 4 =1 Question 3 Every real number is also a complex number? True or false? Answer 1 True. We know that a complex number has two parts: real and imaginary. A complex number is written in form 𝑥 + 𝑦 𝑖 x+yi. Real part is 𝑥 x and imaginary part is 𝑦 y. If the imaginary part is 0, then we have a real number. And if both real and imaginary parts are 0, the number will be 0. We can write any real number as a complex number. Example: We can write 2 as 2 + 0 𝑖 2+0i. Therefore, all real numbers are also complex numbers. Question 4 Determine whether a linear, quadratic, or exponential function best models the data. Then, use regression to find the function that models the data. x 0 1 2 3 4 y 100 89.5 78.9 68.4 57.8 Answer 1 a) Check the 1st differences of the consecutive y-values. Since they are nearly constant, we can use a linear model. y 1st differences 100 89.5 -10.5 78.9 -10.6 68.4 -10.5 57.8 -10.6 b) At the WolframAlpha web site you can enter: fit linear { { 0 , 100 } , { 1 , 89.5 } , { 2 , 78.9 } , { 3 , 68.4 } , { 4 , 57.8 } } {{0,100},{1,89.5},{2,78.9},{3,68.4},{4,57.8}} Which results in 𝑦 = 100.02 − 10.55 𝑥 y=100.02−10.55x Question 5 Determine which type of function—linear, quadratic, or exponential—would be best for modeling each situation below. You do not need to write an equation for the situation. a. Bailey is growing bacteria on a petri dish for science class. The population of bacteria doubles every 20 minutes, so the initial number of bacteria is 1000, then 2000, then 4000, then 8000, etc. b. Avery is working on her math homework. She finished nine problems in five minutes. She knows that, on average, she can do three problems in five minutes. c. Keiton loves catapults. He is building a small catapult that will launch an apple into the air and over the fence into his friend's backyard. Answer 1 a. The function increases very rapidly and thus the best type of function is exponential. b. Since the increase is constant, the best type of function is linear. c. The apple will travel an arc in the shape of a parabola and thus a quadratic function is best.