Ch2.1A – Quadratic Functions Polynomial function of x with degree n: f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Ex: Quadratic function: f(x) = ax2 + bx + c Graphs of quadratic functions are: _____________

Ch2.1A – Quadratic Functions Polynomial function of x with degree n: f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Ex: Quadratic function: f(x) = ax2 + bx + c Graphs of quadratic functions are: parabolas! If a > 0: If a < 0:

Ch2.1A – Quadratic Functions Polynomial function of x with degree n: f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Ex: Quadratic function: f(x) = ax2 + bx + c Graphs of quadratic functions are: parabolas! If a > 0: If a < 0: vertex axis of symmetry (maximum) (minimum)

c) h(x) = –x2 + 1 d) k(x) = (x+2)2 – 3 Ex1) How does each graph compare to y = x2? a) f(x) = b) g(x) = 2x2 c) h(x) = –x2 + 1 d) k(x) = (x+2)2 – 3

c) h(x) = –x2 + 1 d) k(x) = (x+2)2 – 3 Ex1) How does each graph compare to y = x2? a) f(x) = b) g(x) = 2x2 c) h(x) = –x2 + 1 d) k(x) = (x+2)2 – 3 y = ax2 If a > 1 (skinny, up) 0 < a < 1 (wide, up) If a < –1 (skinny, down) –1 < a < 0 (wide, down)

Standard Form of a Quadratic Function: f(x) = a(x – h)2 + k Ex2) Describe the graph of f(x) = 2x2 + 8x + 7

Ex3) Describe the graph of f(x) = –x2 + 6x – 8

HW#14) Describe the graph of f(x) = ½x2 – 4 HW#17) Describe the graph of f(x) = x2 – x + 5/4 HW#20) Describe the graph of f(x) = –x2 – 4x + 1 Ch2.1A p165 13-21odd

Ch2.1A p165 13-21odd

f(x) = a(x – h)2 + k Ch2.1B – Finding Quadratic Functions Ex4) Find the equation for the parabola that has a vertex at (1,2) and passes thru (0,0), as shown.

f(x) = a(x – h)2 + k HW#36) Find the equation for the parabola that has a vertex at (-2,-2) and passes thru (-1,0), as shown. Ch2.1B p166 14-22 even, 31-35 odd

Ch2.1B p166 14-22 even ,31-35 odd

Ch2.1B p166 14-22 even ,31-35 odd

Ch2.1C – Quadratic Word Problems Ex5) The height of a ball thrown can be found using the equation f(x) = –0.0032x2 + x + 3 where f(x) is the height of the ball and x is the distance from where its thrown. Find the maximum height.

Ex6) The percent of income (P) that families give to charity varies with income (x) by the following function: P(x) = 0.0014x2 – 0.1529x + 5.855 5 < x < 100 What income level corresponds to the minimum percent? Ch2.1C p167+ 32,34,36,53,55,57,59

Ch2.1C p167+ 32,34,36,53,55,57,59

R = 900x – 0.1x2 where R is revenue and x is units sold. Ch2.1C p167+ 32,34,36,53,55,57,59 53. Find the max # units that produces a max revenue given by R = 900x – 0.1x2 where R is revenue and x is units sold. 55. A rancher has 200ft of fencing to enclose corrals. Determine the max enclosed area. Write a function. x x A = (2x).y y P = (2x) + (2x) + y + y 200 = x + x + x + x + y + y + y

57. The height y of a ball thrown by a child is given by: x is horiz distance. a. Graph on calc. b. How high when leaves childs hand at x = 0? c. Max height? d. How far when strikes ground? 59. # Board feet (V) as a function of diameter (x) given by: V(x) = 0.77x2 – 1.32x – 9.31 5 < x < 40 a) graph b) estimate # board feet in 16 in diameter log c) Est diam when 500 board feet.

Ch2.2A – Polynomial Functions of Higher Degree Graphs of polynomial functions are always smooth and continuous

Types of simple graphs: y = xn When n is even: When n is odd: Exs: Exs:

Ex1) Sketch: a) f(x) = –x5 b) g(x) = x4 +1 c) h(x) = (x+1)4

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Leading Coefficient Test (An attempt to see where a graph is going.) f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 When n is even: (an > 1) (an < 1) When n is odd: (an > 1) (an < 1)

Ch2.2A p177 1-4,17-26 Ex2) Use LCT to determ behavior of graphs: a) f(x) = –x3 + 4x b) g(x) = x4 – 5x2 + 4 c) h(x) = x4 – x Ch2.2A p177 1-4,17-26

Ch2.2A p177 1-4,17-26

Ch2.2A p177 1-4,17-26

Ch2.2A p177 1-4,17-26

Ch2.2A p177 1-4,17-26

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Ch2.2B – Zeros f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 1. Graph has at most n zeros. 2. Has at most n – 1 relative extrema (bumps on the graph). Ex3) Find all the zeros of f(x) = x3 – x2 – 2x

Ex4) Find all the real zeros of f(x) = x5 – 3x3 – x2 – 4x – 1 Ex5) Find the polymonial with the following zeros: –2, –1, 1, 2 Ch2.2B p178 35 – 55 odd

Ch2.2B p178 35 – 55 odd

Ch2.2B p178 35 – 55 odd

Ex1) Divide f(x) = 6x3 – 19x2 – 4 by (x – 2) then factor completely. Ch2.3 – More Zeros Ex1) Divide f(x) = 6x3 – 19x2 – 4 by (x – 2) then factor completely.

Ex2) Divide f(x) = x3 – 1 by (x – 1)

Ex3) Divide f(x) = 2x4 + 4x3– 5x2 + 3x – 2 by x2 + 2x – 3

Going down, add terms. Going diagonally multiply by the zero. Synthetic Division Going down, add terms. Going diagonally multiply by the zero. Ex4) Divide x4 – 10x2 – 2x + 4 by (x + 3) Ex5) Divide

The Remainder Theorem – if u evaluate (divide) a function for a certain x in the domain, the remainder will equal the corresponding y from the range. Ex5) Evaluate f(x) = 3x3 + 8x2 + 5x – 7 at x = –2 Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

Ch2.3A p191 7–19odd, 23–31odd, 41–47odd

f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 Ch2.3B – Rational Zero Test f(x) = anxn + an-1xn-1 + . . . a2x2 + a1x + a0 any factor any factor of this (q) of this (p) Possible zeros: Ex1) Find all the zeros of f(x) = 4x3 + 4x2 – 7x + 2.

Ex2) Find all the zeros of f(x) = x3 – 10x2 + 27x – 22 Ch2.3B p192 51 – 60 all

Ch2.3B p192 51 – 60 all HW#55) Find all the zeros of f(x) = x3 + x2 – 4x – 4

Ch2.3B p192 51 – 60 all HW#60) Find all the zeros of f(x) = 4x4 – 17x2 + 4

Ch2.3B p192 51 – 60 all

Ch2.3B p192 51 – 60 all

Ch2.3C p192 8-16even, 24-30even,61-69odd 8) Divide 5x2 – 17x – 12 by (x – 4)

Ch2.3C p192 8-16even, 24-30even,61-69odd 16) Divide x3 – 9 by (x2 + 1)

Ch2.3C p192 8-16even, 24-30even,61-69odd 24) Synthetic Divide 9x3 – 16x – 18x2 +32 by (x – 2)

Ch2.3C p192 8-16even, 24-30even,61-69odd 30) Synthetic Divide –3x4 by (x + 2)

Ch2.3C p192 8-16even, 24-30even,61-69odd 61) Zeros: 32x3 – 52x2 + 17x + 3

Ch2.3C p192 8-16even, 24-30even,61-69odd 69) Zeros: 2x4 – 11x3 – 6x2 + 64x + 32 = 0

Ch2.3C p192 8-16even, 24-30even,61-69odd 8,16,24,30,61,69 in class

Ch2.3C p192 8-16even, 24-30even,61-69odd 8,16,24,30,61,69 in class

Ch2.3C p192 8-16even, 24-30even,61-69odd 8,16,24,30,61,69 in class

Ch2.3C p192 8-16even, 24-30even,61-69odd 8,16,24,30,61,69 in class

Ch2.4 – Complex Numbers x2 + 1 = 0

x2 + 1 = 0 Real Imaginary Ch2.4 – Complex Numbers Complex Numbers have the standard form: a + bi Real Imaginary Quick Review: Unit Unit Rational numbers  normal ex: 2.5 Irrational numbers  square roots ex: Imaginary numbers  negative square roots ex:

b) 2i + (–4 – 2i) = c) 3 – (–2 – 3i) + (–5 + i) = b) (2 – i)(4 + 3i) = Ex1) a) (3 – i) + (2 + 3i) = b) 2i + (–4 – 2i) = c) 3 – (–2 – 3i) + (–5 + i) = Ex2) a) (i)(–3i) = b) (2 – i)(4 + 3i) = c) (3 + 2i)(3 – 2i) = complex conjugates  their product is a real #! Important for getting I out of the denominator.

Ex3) Ex4)

Ex5) Plot complex #’s in the complex plane: a) 2 + 3i b) –1 + 2i c) 4 + 0i Imag axis Real axis HW#1) Solve for a and b: a + bi = –10 + 6i HW#5) Solve: Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.4 p202 1–63odd,67–81odd

Ch2.5A – Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, it has at least one zero in the complex plane. Ex1) Write f(x) = x5 + x3 + 2x2 – 12x + 8 as a product of linear factors. Ch2.5A p210 9 – 21 all

HW#9) Write f(x) = x2 + 25 as a product of linear factors. HW#14) f(y) = y4 – 625

HW#15) Write f(z) = z2 – 2z + 2 as a product of linear factors. HW#20) Write f(s) = 2s3 – 5s2 + 12s – 5 Ch2.5A p210 9 – 21 all

Ch2.5A p210 9 – 21 all

Ch2.5B – More FTA If f(x) is a polynomial of degree n, it has at least one zero in the complex plane. Ex2) Write a fourth degree polynomial that has –1, +1, and 3i as zeros.

Ex3) Find all zeros of f(x) = x3 – 4x2 + 9x – 36 if 3i is a zero. Ch2.5B p210 23–35odd, 41-43all

HW#33) i, –i, 6i, –6i 43) Find all zeros of f(x) = 2x4 – x3 + 7x2 – 4x – 4, r = 2i. Ch2.5B p210 23–35odd, 41-43all

Ch2.5B p210 23–35odd, 41-43all

Ch2.5B p210 23–35odd, 41-43all

Ch2.5B p210 23–35odd, 41-43all

Ch2.6 – Rational Functions and Asymptotes Ex1) Find the domain of and what happens near the excluded values of x?

Ch2.6 – Rational Functions and Asymptotes Ex1) Find the domain of and what happens near the excluded values of x? For any function f(x): -If n < m, x axis is a horizontal asymptote -If n > m, no horizontal asymptote -If n = m, the line is a horizontal asymptote

-If n < m, x axis is a horizontal asymptote -If n > m, no horizontal asymptote -If n = m, the line is a horizontal asymptote Ex2) List the horiz asymptotes: a) b) c) Ex3) This non-rational function has 2 horiz asymptotes, to the left and right of x = 0. Find them algebraically and graphically. Ch2.6 p218 1,3,7,11-19odd

Ch2.6 p218 1,3,7,11-19odd

Ch2.6 p218 1,3,7,11-19odd

Ch2.6 p218 1,3,7,11-19odd

Ch2.6 p218 1,3,7,11-19odd

Ch2.7 – Graphs of Rational Functions 1. y-intercept is the value of f(0). 2. x-intercepts are the zeros of the numerator. Solve p(x) = 0. (If any.) 3. Vertical asymptotes are the zeros of the denominator. Solve q(x) = 0. (If any.) (Look for the graph to approach +/– .) 4. Horizontal asymptotes where f(x) increases or decreases without bound. (Approaches but does not reach some #.) (Notes from yesterday.) 5. You’ll have to figure out what’s going on everywhere else. (Don’t forget to take advantage of ur calculator.) Ex1) Analyze the function

Ex1) Analyze the function 1. y-int: 2. x-int: 3. vert asymp: 4. horiz asymp: x g(x) 1 -4 3 5

Ex2) Analyze the function 1. y-int: 2. x-int: 3. vert asymp: 4. horiz asymp: x f(x) 1 10 -1 -10

Ex3) Analyze the function 1. y-int: 2. x-int: 3. vert asymp: 4. horiz asymp: x f(x) Ch2.7A p227 13 – 23odd, 31,33

Ch2.7A p227 13 – 23odd, 31,32

Ch2.7A p227 13 – 23odd, 31,32

Ch2.7B – More Graphing Ex4) Analyze the function 1. y-int: 2. x-int: 3. vert asymp: 4. horiz asymp: x f(x)

Slant asymptotes If the degree of the numerator is exactly one more than the denominator, you get a slant asymptote. Use long division to find it Ex4) Graph 1. y-int: 2. x-int: 3. vert asymp: 4. horiz asymp: 5. slant asymp:

HW#50) Graph 1. y-int: 2. x-int: 3. vert asymp: 4. horiz asymp: 5. slant asymp: Ch2.7B p22749-55odd,50

Ch2.7B p22749-55odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd

Ch2 Rev p232 1-15odd,19-29odd,33,35,41-46all,47-71odd