Question 1 absolute value of a complex number, axis of symmetry, binomial, completing the square, complex conjugate, complex number, complex plane, discriminant, imaginary number, imaginary part, imaginary unit, maximum value, minimum value, parabola, quadratic function, quadratic inequality in two variables, quadratic model, quadratic regression, real part, root of an equation, standard form, trinomial, vertex form, vertex of a parabola, zero of a function. Complete the sentences below with vocabulary words from the list above. The number 5i can be classified as both a(n) _______ and a _______. Answer 1 The number 5i can be classified as both a(n) imaginary number and a complex number. Question 2 The variable z is often used to denote a complex number and 𝑧 ˉ z ˉ is used to denote its conjugate. If 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi, simplify the expression 𝑧 2 − 𝑧 ˉ 2 z 2 − z ˉ 2 . Answer 1 With 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi, the conjugate, 𝑧 ˉ z ˉ , is 𝑎 − 𝑏 𝑖 a−bi. Hence, the expression 𝑧 2 − 𝑧 ˉ 2 z 2 − z ˉ 2 is equivalent to ( 𝑎 + 𝑏 𝑖 ) 2 − ( 𝑎 − 𝑏 𝑖 ) 2 = [ 𝑎 2 + 2 𝑎 ( 𝑏 𝑖 ) + ( 𝑏 𝑖 ) 2 ] − [ 𝑎 2 − 2 𝑎 ( 𝑏 𝑖 ) + ( 𝑏 𝑖 ) 2 ] (a+bi) 2 −(a−bi) 2 =[a 2 +2a(bi)+(bi) 2 ]−[a 2 −2a(bi)+(bi) 2 ] = 𝑎 2 + 2 𝑎 𝑏 𝑖 − 𝑏 2 − ( 𝑎 2 − 2 𝑎 𝑏 𝑖 − 𝑏 2 ) = 4 𝑎 𝑏 𝑖 =a 2 +2abi−b 2 −(a 2 −2abi−b 2 )=4abi Hence, 𝑧 2 − 𝑧 ˉ 2 = ( 4 𝑎 𝑏 𝑖 ) 𝑖 z 2 − z ˉ 2 =(4abi)i. Question 3 What is the complex plane? How do we graph a complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi in the complex plane? Answer 1 The complex plane is defined as a 2-dimensional plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number. A complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi is graphed by plotting the point (a, b) in this plane. Question 4 What are the modulus and argument of the complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi? Answer 1 The modulus of a complex number 𝑧 = 𝑎 + 𝑏 𝑖 z=a+bi is given by 𝑟 = 𝑎 2 + 𝑏 2 r= a 2 +b 2 , and the argument 𝜃 θ is defined as 𝜃 = tan ⁡ − 1 ( 𝑏 𝑎 ) θ=tan −1 ( a b ). In the complex plane, this corresponds to the distance from the origin to the point (a, b) and the angle formed with the positive real axis. Question 5 Write a compound inequality that represents all real numbers that are no more than 1 or no less than 6. Answer 1 No more than 1, meaning less than or equal to 1, is mathematically represented as: 𝑥 ≤ 1 x≤1 Greater than or equal to 6 is represented as: 𝑥 ≥ 6 x≥6 So, the compound inequality is: 𝑥 ≤ 1 x≤1 OR 𝑥 ≥ 6 x≥6