Pre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers

Transcription

1 0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex numbers Complex number: ones in standard form a + bi Real part: a Imaginary part: b When a 0 and b = 0, then a + 0i = a (just the real number) When b 0 the complex number is an imaginary number When a = 0 and b 0, then 0 + bi is a pure imaginary number Sep 2 10:41 AM i = i 2 = 1 i 3 = i i 4 = 1 = i Simplify. 1. i 7 = 2. i 20 = 3. i 55 = 4. i 93 = 4. i 93 = 5. i 13 = Sep 2 11:03 AM 1

2 Addition: combine like terms, respective of the signs Simplify. 6. (4 i) + ( 3 + 5i) 7. (6 2i) + ( 5 3i) Subtraction: first distribute the negative, then combine like terms, respective of the signs Simplify. 8. (8 5i) ( 4 2i) 9. (3 2i) (5 i) Sep 2 11:16 AM Multiplication: two complex numbers, FOIL real and complex, distribute pure imaginary and complex, distribute Simplify. 10. (2 3i)(6 + 7i) 11. (3 2i)(4 7i) 12. (3 2i) (4 6i) i(2 3i) Sep 2 11:34 AM 2

3 Division: Write as a fraction, then use conjugates. Simplify. 15. (12 + 3i) (1 + 2i) 16. (4 5i) (1 2i) Sep 2 11:41 AM Solve equations by taking the square root of each side x = x = 16 Sep 3 10:47 AM 3

4 Find the values of x and y to make each equation true x iy = 8 + 7i 20. 5x + 3iy = 5 6i Sep 3 10:51 AM 2 1 Power and Radical Functions Objectives: To graph and analyze power functions To graph and analyze radical functions and solve radical equations Power function: any function of the form f(x) = ax n, where a and n are nonzero constant real numbers. Monomial function: any function that can be written as f(x) = a or f(x) = ax n, where a and n are nonzero constant real numbers. Aug 30 8:33 AM 4

5 Example 1: Aug 30 9:22 AM Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and Aug 11 3:01 PM 5

6 Jan 22 8:03 AM Functions with negative exponents: Recall x 1 = 1/x, thus it is undefined at x = 0, and there will be discontinuities in these graphs. When the exponent is even, the end behaviors When the exponent is odd, the end behaviors at the vertical asymptote will both go to at the vertical asymptote will go in positive infinity or negative infinity. opposite directions. f(x) = 3x 2 f(x) = 3/4 x 5 Example 2: Aug 30 9:04 AM 6

7 7. 8. Aug 11 3:03 PM Rational Exponents: Recall that indicates the nth root of x, and, where p/n is in simplest form, indicates the nth root of x p. If n is an even integer, then the domain must be restricted to nonnegative values. Functions with exponents that have fractions in simplest form. When the denominator is even, there will be a restriction on the domain since it cannot be negative under the square root. When the denominator is odd, there will not be a restriction on the domain. Example 3: Jan 1 1:18 PM 7

8 Aug 11 3:12 PM Radical function: a function that can be written as than 1 that have no common factors., where n and p are positive integers greater Aug 31 9:03 AM 8

9 Example 5: Aug 31 9:11 AM Solve radical equations: 1. Isolate the radical, if possible 2. Raise each side of the equation to a power equal to the index of the radical 3. Solve the resulting equation 4. Check for extraneous roots (raising each side of an equation to a power sometimes produces solutions that do not satisfy the original equation). Example 6: Solve each equation Aug 31 9:03 AM 9

10 Solve each equation Aug 31 9:17 AM 10

11 Attachments notebook 21881e912c2.galleryitem