ch2.1a – quadratic functions polynomial function of x with degree n: f(x) = a n x n + a n-1 x n-1...

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: _____________





Graphs of quadratic functions are: parabolas!

If a > 0: If a < 0:





Graphs of quadratic functions are: parabolas!

If a > 0: If a < 0:

vertexaxis of symmetry (maximum)

vertex (minimum)

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

2

3

1x

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)

2

3

1x

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

Ch2.1B – Finding Quadratic Functions

f(x) = a(x – h)2 + k

Ex4) Find the equation for the parabola that has a vertex at (1,2) andpasses 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) andpasses thru (-1,0), as shown.

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 + 3where 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 < 100What 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

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 xA = (2x).y

yP = (2x) + (2x) + y + y

200 = x + x + x + x + y + y + y

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

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 < 40a) graphb) estimate # board feet in 16 in diameter logc) Est diam when 500 board feet.

4212

1 2 xxy

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

Leading Coefficient Test (An attempt to see where a graph is going.)


When n is even: (an > 1) (an < 1)

When n is odd: (an > 1) (an < 1)

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

Ch2.2B – Zeros


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

Ch2.3 – 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

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

Ch2.3B – Rational Zero Test


any factor any factorof this (q) of this (p)

Possible zeros:

Ex1) Find all the zeros of f(x) = 4x3 + 4x2 – 7x + 2.

q

p

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

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

8) Divide 5x2 – 17x – 12 by (x – 4)


16) Divide x3 – 9 by (x2 + 1)


24) Synthetic Divide 9x3 – 16x – 18x2 +32 by (x – 2)


30) Synthetic Divide –3x4 by (x + 2)


61) Zeros: 32x3 – 52x2 + 17x + 3


69) Zeros: 2x4 – 11x3 – 6x2 + 64x + 32 = 0

Ch2.3C p192 8-16even, 24-30even,61-69odd8,16,24,30,61,69 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

Ch2.4 – Complex Numbers

x2 + 1 = 0

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:

1 or 1

1

2

ii

x

33

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)

i1

1

i

i

24

32

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

94

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 + 2as a product of linear factors.

HW#20) Write f(s) = 2s3 – 5s2 + 12s – 5 as a product of linear factors.

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

Ch2.6 – Rational Functions and Asymptotes

Ex1) Find the domain of and what happens near the excluded values of x? x

xf1

)(

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

xxf

1)(

01

01

...

...)(

bxbxb

axaxaxf

mm

nn

m

n

b

ay

-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

m

n

b

ay

13

2)(

2 x

xxf

13

2)(

2

2

x

xxf

13

2)(

2

3

x

xxf

2

10)(

x

xxf

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

)(

)()(

xq

xpxf

2

3)(

x

xg


1. y-int:

2. x-int:

3. vert asymp:

4. horiz asymp:

x g(x) 0 1 -4 3 5

2

3)(

x

xg


1. y-int:

2. x-int:

3. vert asymp:

4. horiz asymp:

x f(x) 1 10 -1 -10

x

xxf

12)(


1. y-int:

2. x-int:

3. vert asymp:

4. horiz asymp:

x f(x)

Ch2.7A p227 13 – 23odd, 31,33

2)(

2

xx

xxf

Ch2.7A p227 13 – 23odd, 31,32

Ch2.7B – More GraphingEx4) Analyze the function

1. y-int:

2. x-int:

3. vert asymp:

4. horiz asymp:

x f(x)

4

)9(2)(

2

2

x

xxf

Slant asymptotesIf 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:

1

2)(

2

x

xxxf

HW#50) Graph

1. y-int:2. x-int:3. vert asymp:4. horiz asymp:5. slant asymp:

Ch2.7B p22749-55odd,50

x

xxf

21)(

Ch2.7B p22749-55odd

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

ch2.1a – quadratic functions polynomial function of x with degree n: f(x) = a n x n + a n-1 x n-1...

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