Exploring Quadratic Equations and Complex Numbers
Dive into a comprehensive review of quadratic equations through interactive stations, graph analysis, and problem-solving exercises. Learn about parabolic paths, completing the square, and the use of the quadratic formula. Explore the concepts of complex numbers, imaginary units, and quadratic functions in a structured and engaging format.
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Quadratic Review Stations Start with 6 students at each station. Work for 4 minutes. Check answers for 2 minutes. Move clockwise to the next station. Timer
by Factoring. (?? ?)(? + ?) ?(?? ?) ?(? ?)(? + ?)
Station 5 A baseball player tries to hit a ball over an outfield fence that is 4 m high and 110 m from home plate. The ball is hit 1 m above home plate and reaches its highest point 30 m above a point on the ground that is 60 m from home plate. 15. Make a sketch showing the path of the baseball.
Station 5 A baseball player tries to hit a ball over an outfield fence that is 4 m high and 110 m from home plate. The ball is hit 1 m above home plate and reaches its highest point 30 m above a point on the ground that is 60 m from home plate. 16. If home plate is at the origin of the coordinate system, find an equation of the parabolic path of the baseball. ?? ????? ???+ ?? ? = 17. Will the ball go over the outfield fence? yes
Chapter 1 - Sections 5 through 8 *Solving a Quadratic Equation ??2+ ?? + ? = 0 *Graphing a Quadratic Function *Quadratic Models *Complex Numbers Key Terms Imaginary Unit i Complex Number Complex Conjugates Quadratic Equation Quadratic Model Main Ideas Completing the Square Discriminant Quadratic Function
Complex Numbers Any number of the form a+bi, where a and b are real numbers and i is the imaginary unit is called a complex number. a is the called the real part and b is called the imaginary part. Example: 6 7?
Imaginary Unit The imaginary unit is i which has the following properties: ? ?2= 1 = 1 i = and i 3 ? 1 ? = = 1 i i 2 4 = i 5
Square Root of Negative Numbers if = 36 6? 0 a = 11 ? 11 = = a i a 24 ? 24 = ? 4 6 = ? 4 6 = 2? 6
Complex Numbers 1.? ? 2. ?? 3. ?? ? 4. ? ?? ? + ? 5. ? + ?? ? ? 6. ? + ? ? -1 Simplify: ? ?? = ? ?? ? = ?? ? ?
*Complex Numbers Main Ideas To simplify an expression that contains the square root of a negative number, begin by writing the expression in terms of i. ?? ? = ? ?? ? ? = ? ? ? ? ? = ?? ? = ?? ? ? ? To add or subtract complex numbers, group the real parts and the imaginary parts. ? ?? ? + ? = ? ?? + ? ? = ? ??
*Complex Numbers Main Ideas To multiply two complex numbers use FOIL (first, outer, inner, last) and substitute 1 for ?2. ? + ?? ? ? = ?? ?? + ??? ??? = ?? + ??? ? ? = ?? + ???
*Complex Numbers Main Ideas To simplify an expression with a complex number in the denominator, multiply the numerator and denominator by the complex conjugate of the denominator. ? = ? ? ? ? + ? ? ? ? ? ? =? ? ? ? ??? ? + ? ? =? ? =? ? ? ? ? ? ? ? ? ? ?? = ? Online Notes & Practice for Complex Numbers
Main Ideas *Quadratic Models Homework #14 Or use quadratic regression on the calculator! To enter the data: Press [STAT] [1] to access the STAT list editor. Input the data in the L1 and L2 lists, pressing [ENTER] after each number. Press [2nd] [MODE] to QUIT and return to the home screen. To calculate the quadratic regression (ax2+bx+c): Press [STAT] to access the STAT menu. Scroll right to highlight the CALC menu. Press [5] to select QuadReg(ax+b). *Press enter 5 times or: Press [2nd] [1] [,] [2nd] [2] to input L1,L2. Press [,] [VARS], scroll right to highlight Y-VARS, then press [1] [1] to input ,Y1. Press [ENTER] to calculate the quadratic regression.
Homework Sections 1.5 1.8 Worksheet
