Ch2.1A – Quadratic Functions
Polynomial function of x with degree n:
f(x) = anxn + an-1xn-1 +
Ex: Quadratic function:
.. .
a2x2 + a1x + a0
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 +
Ex: Quadratic function:
.. .
a2x2 + a1x + a0
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 +
Ex: Quadratic function:
.. .
a2x2 + a1x + a0
f(x) = ax2 + bx + c
Graphs of quadratic functions are: parabolas!
If a > 0:
If a < 0:
axis of symmetry
vertex
(minimum)
vertex
(maximum)
Ex1) How does each graph compare to y
= x2?
1 2
a) f(x) = x
3
b) g(x) = 2x2
c) h(x) = –x2 + 1
d) k(x) = (x+2)2 – 3
= x2?
b) g(x) = 2x2
Ex1) How does each graph compare to y
1 2
a) f(x) =
3
x
c) h(x) = –x2 + 1
d) k(x) = (x+2)2 – 3
y = ax2
If a > 1 (skinny, up)
If a < –1 (skinny, down)
0 < a < 1 (wide, up)
–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
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) 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
What income level corresponds to the minimum percent?
Ch2.1C p167+ 32,34,36,53,55,57,59
5 < x < 100
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
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:
1 2
y   x  2x  4
12
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
a) graph
b) estimate # board feet in 16 in diameter log
c) Est diam when 500 board feet.
5 < x < 40
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:
Exs:
When n is odd:
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.)
f(x) = anxn + an-1xn-1 +
When n is even:
When n is odd:
(an > 1)
(an > 1)
.. .
a2x2 + a1x + a0
(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
Ch2.2A p177 1-4,17-26
Ch2.2A p177 1-4,17-26
Ch2.2A p177 1-4,17-26
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
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
Ch2.3A p191 7–19odd, 23–31odd, 41–47odd
Ch2.3A p191 7–19odd, 23–31odd, 41–47odd
Ch2.3B – Rational Zero Test
f(x) = anxn + an-1xn-1 +
any factor
of this (q)
.. .
a2x2 + a1x + a0
any factor
of this (p)
p
Possible zeros:
q
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
Ch2.4 – Complex Numbers
x2 + 1 = 0
x  1
i 2  1 or i   1
Complex Numbers have the standard form: a + bi
Real
Unit
Imaginary
Unit
Quick Review:
Rational numbers  normal ex: 2.5
Irrational numbers  square roots ex: 3
Imaginary numbers  negative square roots ex:
3
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)
1
1 i
Ex4)
2  3i
4  2i
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:
4 9
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
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
Ch2.5B p210 23–35odd, 41-43all
Ch2.5B p210 23–35odd, 41-43all
Ch2.6 – Rational Functions and Asymptotes
1
Ex1) Find the domain of f ( x )  and what happens near
x
the excluded values of x?
Ch2.6 – Rational Functions and Asymptotes
1
Ex1) Find the domain of f ( x )  and what happens near
x
the excluded values of x?
For any function f(x):
f ( x) 
an x n  ...  a1 x  a0
bm x m  ...  b1 x  b0
-If n < m, x axis is a horizontal asymptote
-If n > m, no horizontal asymptote
an
-If n = m, the line y 
is a horizontal asymptote
bm
-If n < m, x axis is a horizontal asymptote
-If n > m, no horizontal asymptote
an
-If n = m, the line y 
is a horizontal asymptote
bm
Ex2) List the horiz asymptotes:
a) f ( x ) 
2x
3x  1
2
b) f ( x) 
2x2
3x 2  1
c) f ( x) 
2x3
3x 2  1
Ex3) This non-rational function has 2 horiz asymptotes,
to the left and right of x = 0. Find them algebraically
and graphically.
x  10
f ( x) 
Ch2.6 p218 1,3,7,11-19odd
x 2
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
p( x)
f ( x) 
q( x)
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.)
3
Ex1) Analyze the function g ( x) 
x2
.)

3
Ex1) Analyze the function g ( x) 
x2
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x g(x)
0
1
-4
3
5
2x 1
Ex2) Analyze the function f ( x) 
x
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x f(x)
1
10
-1
-10
Ex3) Analyze the function f ( x ) 
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
x f(x)
Ch2.7A p227 13 – 23odd, 31,33
x
x2  x  2
Ch2.7A p227 13 – 23odd, 31,32
Ch2.7A p227 13 – 23odd, 31,32
Ch2.7B – More Graphing
2( x 2  9)
Ex4) Analyze the function f ( x) 
x2  4
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
x2  x  2
Ex4) Graph f ( x) 
x 1
1. y-int:
2. x-int:
3. vert asymp:
4. horiz asymp:
5. slant asymp:
HW#50) Graph
1 x2
f ( x) 
x
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