Warm Up Oct 8 10:36 AM Oct 5 2:22 PM Linear Function Qualities Oct 8 9:22 AM Oct 8 9:19 AM Quadratic Function Qualities Oct 8 9:26 AM Oct 8 9:25 AM 1
Oct 8 9:28 AM Oct 8 9:25 AM Given vertex (-1,4) and point (1,2), write the equation for the quadratic function. Oct 8 9:37 AM Oct 8 9:23 AM Vertical Free Fall Motion Homework: Page 169 #1-12 odds, 13-18, 23-44 odds Oct 8 10:08 AM Oct 8 9:26 AM 2
Homework: Page 170 # 45-48, 51, 54-59, 61-63, 69, 70 Example: and finish homework from pg 169. Given: f(x) = x 2 +3 Find the average rate of change from x = 3 to x = 6. Oct 8 9:22 AM Oct 8 10:12 AM Warm-Up Oct 8 10:42 AM Oct 8 10:44 AM Example 1: Example 2: State whether the following functions are power functions. If yes, give the power and leading coefficient. a. f(x) = 16x 2/3 b. f(x) = 5(2 x ) c. f(x) = 8ax 6 Oct 8 10:46 AM Oct 8 10:44 AM 3
Example: State whether the following are monomial functions or not. If yes, state the degree and leading coefficient. If not, say why not. a. f(x) = 4 x 8 b. y = 5x -3 c. -7(6 x ) 3 Oct 8 10:44 AM Oct 8 10:45 AM Examples: Write the statement as a power function equation. USe k for the constant of variation if one is not given. a. The volume of a circular cylinder with fixed height is proportional to the square of its radius r. Homework: Page 182 #1-22 odds, 37-42 all, 53-57 odd, 58-62 all b. Charles Law states that the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute temperature T. c. Distance d varies inversely with time t and has a constant of variation of k = velocity v. d. The speed p of a falling object that has been dropped from rest varies as the square root of the distance d traveled, with a constant of variation k = 2g Oct 13 8:57 AM Oct 8 10:49 AM Section 2.3 Polynomial Functions of Higher Degree with Modeling Recall : the degree of a polynomial is the highest power. The leading coefficient is the number attached to the highest power term. where a 3 is the leading coefficient, if: Oct 17 8:44 PM Oct 17 8:47 PM 4
where a 4 is the leading coefficient, if: Oct 17 8:47 PM Oct 17 8:48 PM aka: end behavior is determined by the leading coefficient and degree of the polynomial. Make chart of even vs odd to know end behavior Oct 17 8:48 PM Oct 17 8:49 PM Better Explained as: If there is a power on the outside of the linear factor for the zero, then the zero has multiplicity. If a zero has odd multiplicity, the graph of the function crosses the x-axis at that zero. If a zero has even multiplicity, the graph of the function touches the x-axis at that zero but does not go through the x-axis. Oct 17 8:49 PM Oct 17 9:00 PM 5
Example: Graph the function in a viewing window that shows all of its extrema and x-ints. Describe end behavior with limits. f(x) = (2x-3)(4-x)(x+1) w/ y-int of 8 f(x) = -3x 4-5x 3-17x 2 + 14x + 41 plot y int and zeroes first, then figure out end behavior, then look at multiplicities with zeroes to connect inside in graph together. Oct 17 8:59 PM Oct 17 9:12 PM Homework: Page 193 #1-7 odds, 9-12, 33-42 odd, 67, 69-73 Hint: when matching an equation with a graph without a calculator- look at leading coefficient and degree of the polynomial first. **In practice, the IVT is used in combination with our other math knowledge to explain or prove statements. See Ex 7 page 190 Oct 17 8:58 PM Oct 17 9:24 PM 2.4 Real Zeros of Polynomial Functions Example: Use syntheic division to divide 2x 3-3x 2-5x - 12 by x - 3. Example: Use long division to find the quotient and remainder when 2x 4 - x 3-2 is divided by 2x 2 + x + 1. Write your answer in polynomial and fractional form. 3 2-3 -5-12 2 2x 4 - x 3-2 2x 2 + x + 1 fractional form = x 2 x + x+2 2x 2 + x +1 polynomial form = 2x 3 3x 2 5x 12 Oct 18 9:55 PM Oct 18 9:59 PM 6
Remainder Theorem: If a polynomial f(x) is divided by x - k then the remainder is r = f(k). Factor Theorem: A polynomial function f(x) has a factor x-k if and only if f(k) = 0. Example: Find the remainder when f(x) = 3x 2 + 7x - 20 is divided by: a. x 2 b. x + 1 c. x + 4 f( 4) = 3( 4) 2 + 7( 4) 20 48 28 20 = 0 Example: Show that x+4 is a factor of 3x 2 + 7x - 20 x+4 3x 2 + 7x - 20 OR f(2)= 3(2) 2 + 7(2) 20 12 + 14 20 = 6 OR f ( 1) = 3( 1) 2 + 7( 1) 20 3 7 20 = 24 Oct 18 10:10 PM Oct 18 10:12 PM the one zero we can definitely find in our table Note: Use synthetic division to confirm potential zeroes. this yields: 2x 2-4 (factoring out a 2 gives): Oct 18 10:04 PM Oct 18 10:04 PM See page 202 with Upper & Lower bounds and Ex 6 on page 203 Homework: Page 205 # 1-23 odd Section 2.4 (Part 2) p. 205 25 35 odd 49 55 odd 57 60, 65 68 Oct 18 10:07 PM Oct 19 9:27 AM 7
Section 2.5 Complex Zeroes & The Fundamental Theorem of Algebra Fundamental Theorem of Algebra: A polynomial of degree n has n complex zeroes (real & nonreal). Example: Write the polynomial function in standard form and identify the zeroes and x-intercepts. a. f(x)= (x - 2i)(x + 2i) b. f(x) = (x - 3)(x - 3)(x - i)(x + i) Oct 19 9:16 AM Oct 19 9:22 AM Example: Write a polynomial function of minimum degree in standard form with real coefficients whose zeroes include those listed. Zeroes are 2, -3, and 1 + i. Example: Write a polynomial function of minimum degree in standard form with real coefficients whose zeroes and their multiplicities include those listed. Given: -1 with multiplicity 3 and 3 with multiplicity 1 Oct 19 9:23 AM Oct 19 9:29 AM Homework: Page 215 #2, 9, 12, 15, 17-20, 27, 29, 31, 34, 37, 39 Oct 19 9:27 AM Oct 19 9:27 AM 8
Section 2.6 Graphs of Rational Functions Oct 19 10:57 AM Oct 19 11:04 AM Example: Given the graph below, evaluate the limits. a. lim f(x) x -1 + b. lim f(x) x -1 - Graphs of Rational Functions x-int: occur when the numerator = 0 (but it cannot also make the den = 0) y-int: occur when the value of f(0) is defined (remember x = 0 gives the y-int) c. lim f(x) x - Asymptotes Information V. A. occur when the den = 0. d. lim f(x) x + H.A. occur when the 1: (degree of the num) = (deg of den) -in this case, the H.A. is "y = ratio of the leading coefficients of num. & den." 2: (degree of den.) is larger than (degree of num.) -in this case, the H.A. is y = 0 Oct 20 10:45 AM Oct 19 11:05 AM Example: x-int: y-int: V.A. H.A. In Summary: Oct 19 11:09 AM Oct 19 11:06 AM 9
**You do NOT have to graph the functions as stated in the directions- but do need to find all the information it is asking for. p. 225 1 10 mod 3, 11 14 23, 24, 27, 28 31 36 41, 42, 53 65 68 Oct 20 10:42 AM Oct 21 11:00 AM Solving Equations in One Variable 3 creates a "0" on the bottom of the first fraction, and cant divide by 0 in math. Oct 19 10:58 AM Oct 21 11:00 AM Homework: Page 232 #1-18, 23-30 mod 3 p. 232 31, 32, 33, 35 37, 42 Oct 21 11:03 AM Oct 21 11:03 AM 10
Section 2.8 Solving Inequalities in One Variable factor Oct 19 10:58 AM Oct 21 11:11 AM *Make a sign chart Oct 21 11:16 AM Oct 21 11:12 AM Oct 21 11:12 AM Oct 21 11:13 AM 11
Absolute value ex from pg 240 Example: Solving an inequality involving absolute value Solve: x - 2 x + 3 0 because abs. value of x + 3 is in the denominator, f(x) is undefined at x = -3. The only zero is 2. p. 242 7 23 odd 56 61 Section 2.8 (Part 2) p. 242 33 50 mod 3-3 2 Solution: (-,-3) (-3, 2] Oct 21 11:21 AM Oct 21 11:16 AM 12