Honors Pre-Calculus
Chapter 3
Polynomial and Rational Functions
Ms. Carey
Chapter 3 – Assignment Guide
10/9 W: p. 189
TH: p. 211
1-8, 25-35 o, 45, 47, 65 - 73 eoo, 87
11-17 o, 19-29 o (a&b only), 33-53 eoo, 65-67
F: QUIZ 3.1,3.3 (Homecoming)
10/14 M: NO SCHOOL
T: p. 1028
1, 5, 9, 13, 27-33 o, 37-43 o
W: p. 257
1 - 9 o, 63-66 all, 71 (PSAT)
TH: p. 224
1-17 eoo, 23-31 o, 35, 39-45 o
F: p. 234
10/21 M: p. 234
1,3,5,9,11,19
23, 27-33 o, 39
T: QUIZ 3.7,3.4,3.5 (last grade of qtr)
W: p. 243 1 - 29 eoo, 37 - 53 eoo
TH: Review
F: NO SCHOOL (End of 3rd quarter)
10/28 M: Review
T: Ch 3 Test
W: Qtr Review
TH: QTR EXAM
F: Start Ch 4
Honors Pre-Calculus
3.1
Quadratic Functions & Models
Learning Targets: Students will be able to graph a quadratic function, identify the vertex and axis of
symmetry, determine the maximum or minimum value and use it to solve applied problems, and find the
quadratic function of best fit to data.
.
Quadratic function – vertical parabola
f ( x)  ax 2  bx  c where a, b, & c are real numbers, and a  0
f ( x)  a  x  h   k
2
How are a, b, and c related to h and k?
Vertex @ (h,k)
Axis of symmetry: x = h
The effects of ‘a’ on the graph:
If a  1
If 0  a  1
If a  0
If a  0
Vertex? Axis of symmetry? Graph.
28. f ( x)   x 2  4 x
Discriminant behavior
32. f ( x)  x 2  2 x  3
Quadratic equation
ax 2  bx  c  0
b 2  4ac  0
2 real solutions
b 2  4ac  0
1 real solution
b 2  4ac  0
2 complex solutions
Quadratic function
f ( x)  ax 2  bx  c
2 distinct x-intercepts
Graph crosses x-axis twice
one-x-intercept
Graph touches x-axis at its vertex
no x-intercept
Does not cross/touch x-axis
48. Determine the quadratic function whose graph is given.
Vertex: (-2, 6)
(-4, -2)
Enclosing a Rectangular Field
74. Beth has 3000 feet of fencing available to enclose a rectangular area.
(a) Express the area A of the rectangle as a function of the width x of the rectangle.
(b) For what value of x is the area largest?
(c) What is the maximum area?
Height of a ball
88. A physicist throws a ball at an inclination of 45 to the horizontal. The data below represents the height h of
the ball at the instant it has traveled x feet horizontally.
Distance, Height,
x
h
20
25
40
40
60
55
80
65
100
71
120
77
140
77
160
75
180
71
200
64
a. Draw a scatter plot. What kind of relationship do the two variables appear to have?
b. Use a graphing calculator to find the function of best fit for the data.
c. Use the function found in (b) to determine how far the ball will travel before it reaches it
maximum height.
d. Graph function on calculator.
Honors Pre-Calculus
3.3
Polynomial Functions and Models
Learning Targets: Students will be able to identify polynomial functions and their degree, identify zeros and their
multiplicity, and analyze the graph of a polynomial function.
f ( x)  x3
f ( x)  x 2
On calculator: f ( x )  x 5
f ( x)  x 4
f ( x)  x 7
f ( x)  x 6
f ( x)   x3
f ( x)   x 2
f ( x)   x5
f ( x)   x 4
f ( x)   x 7
f ( x)   x 6
Remember: f ( x)   x  h   k
m
Ex 1: Graph f ( x)    x  1  3 using the appropriate transformations.
3
Parent function
f(x)
A polynomial function is a function of the form f ( x)  an x n  an 1 x n 1  ...  a1 x  a0 where an , an1 ,..., a1 , a0 are
real numbers and n is a nonnegative integer.
If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or a root
of f. If r is a zero of f, then
a. r is an x-intercept of the graph of f
b. (x – r) is a factor of f
If the same factor (x – r) occurs more than once, then r is called a repeated, or multiple zero of f.
The highest power of the factor (x – r) is called the multiplicity of the zero r.
Ex. 2:
Determine the zeros and their multiplicities of f ( x)  x 2  x  1 x  3
ODD DEGREE MULTIPLICITIES ______________
EVEN DEGREE MULTIPLICITIES _____________
3
Form a polynomial whose zeros and degree are given.
14. Zeros: -4, 0, 2; degree 3
18. -2, multiplicity 2; 4, multiplicity 2; degree 4
2
1
3

24. For the function f ( x)   x    x  1
3

a. list each real zero and its multiplicity
b. determine whether the graph crosses or touches the x-axis at each x-intercept
c. find the power function that the graph of f resembles for large values of x
A graph of a polynomial function must include:
1. zero(s) (x-intercepts)
2. y-intercept
3. end behavior
4. graph
Use the 4-step process to graph the following polynomial functions.
36. f ( x)  5 x  x  3
4
44. f ( x)  2  x  1  x  3 x  1
2
3

Honors Pre-Calculus
A.5
Polynomial Division and Synthetic Division
Learning Targets: Students will be able to divide polynomials using long division and divide polynomials using
synthetic division.
Remember Long Division:
123917

2)


14)

x 4  1 dividing by x  1


Synthetic Division - Used when divisor is (x-c)
f (x)  x 3  2x 2  3x  1 divided by g(x)  x  1
28.


Ex.
f (x)  x 5  5x 3  10

Ex.

x  2 3x 3  x 2  x  2
3x 3  x 2  x  2 divided by x  2
is x-c a factor?
f (x)  4 x 4  15x 2  4;
x-2

g(x) = x -1
Honors Pre-Calculus
3.7
Real Zeros of Polynomial Functions
Learning Targets: Students will be able to use the remainder and factor theorems and use the Intermediate Value
Theorem.
Remainder Theorem: If f(x) is divided by (x – c), then the remainder is f(c).
Factor Theorem: (x – c) is a factor of f(x) if and only if f(c) = 0.
*** If f(x) is divided by (x – c) and the remainder = 0, then f(c)= 0, (x – c) is a factor, and c is a root.
6. Is (x + 3) a factor of f ( x)  2 x6  18x 4  x 2  9 ?
***A polynomial function of degree n  1 has at most n distinct zeros (remember multiplicity).
Intermediate Value Theorem (IVT)
If f(x) is a polynomial function with real coefficients, and if for real numbers a and b such that f(a) and f(b)
have opposite signs, then there exists at least one zero (root) between a and b.
OR
64. Use IVT to show that f has a zero in the given interval.
f ( x)  x4  8x3  x2  2
1,0
Honors Pre-Calculus
3.4
Rational Functions
Learning Targets: Students will be able to find the domain of a rational function, determine the vertical asymptotes of a
rational function, and determine the horizontal or oblique asymptotes of a rational function.
What is a rational function?
6. What is the domain? Q( x) 
 x 1  x 
3x 2  5 x  2
14. Given the graph, find the
Domain:
Range:
x-intercepts:
y-intercept:
Horizontal asymptote:
Vertical asymptote:
Oblique asymptote:
What do the reciprocal functions look like?
1
y
x
y
1
x2
Using transformations, graph the following.
G ( x) 
2
 x  2
2
R ( x) 
1
3
x 1
R ( x) 
x4
x
Finding asymptotes of rational functions R( x) 
p ( x)
q ( x)
To Find Vertical Asymptotes: Set denominator = 0 and solve. (Your graph will NEVER cross these!)
To Find Horizontal or Oblique Asymptotes: (graphs can cross these, but will backtrack and get close)
A) If deg numerator  deg denominator then y  0 is a horizontal asymptote.
a
B) If deg numerator  deg denominator then y  is a horizontal asymptote.
b
C) If deg numerator  1  deg denominator then there is an oblique asymptote. To find this, you take the
numerator divided by the denominator, drop the remainder, and set = y.
D) If deg numerator  2  deg denominator or more, then we will not do these!
Find all asymptotes.
36. R ( x) 
3x  5
x6
38. G ( x) 
 x2  1
x5
42. F ( x) 
2 x 2  1
2 x3  4 x 2
Honors PreCalculus
3.5
Rational Function II: Analyzing Graphs
Learning Targets: Students will be able to analyze the graph of a rational function.
To graph rational functions that are in simplest form (6 step process):
1. Find the vertical asymptotes: set denominator = 0 and solve
2. Find x-intercept: set numerator = 0 and solve
3. Find y-intercept: let x = 0, and evaluate
4. Find any horizontal/oblique asymptotes (graph can cross these, but will back track and get close):
a) If deg numerator  deg denominator then y  0 is a horizontal asymptote.
a
b) If deg numerator  deg denominator then y  is a horizontal asymptote.
b
c) If deg numerator  1  deg denominator then there is an oblique asymptote. To find this, you take the numerator
divided by the denominator, drop the remainder, and set = y.
d) If deg numerator  2  deg denominator or more, then we will not do these!
5.
Will graph cross the horizontal or oblique asymptote?
a) Find the points of intersection, if any, of the original function and asymptote found in step 4.
b) In other words, let f(x) = #4, and solve.
6.
Plot points (using a table of values) between asymptotes to graph the function.
Graph.
12. R( x) 
1.
2.
3.
4.
5.
6.
x
 x  1 x  2
4.
R( x) 
2x  4
x 1
1.
2.
3.
4.
5.
6.
x 2  3x  2
20. F ( x) 
x 1
1.
2.
3.
4.
5.
6.
x3  1
10. G ( x)  2
x  2x
1.
2.
3.
4.
5.
6.
Honors Pre-Calculus
3.5 Day 2
Rational Function II: Analyzing Graphs cont.
Learning Targets: Students will be able to analyze the graph of a rational function.
R( x) 
26.
 x  1 x  2  x  3
2
x  x  4
1. VA:
2.
Zeros:
3.
y-int:
4. HA/OA
5. Cross?
6. Plot points
x 2  3x  10
R( x)  2
x  8 x  15
28.
1. VA:
2.
Zeros:
3.
y-int:
4. HA/OA
5. Cross?
6. Plot points
R( x) 
32.
x 2  x  30
x6
1. VA:
2.
Zeros:
3.
y-int:
4. HA/OA
5. Cross?
6. Plot points
R( x)  2 x 
34.
9
x
1. VA:
2.
Zeros:
3.
y-int:
4. HA/OA
5. Cross?
6. Plot points
Honors Pre-Calculus
3.6
Rational Inequalities
Learning Targets: Students will be able to solve rational inequalities algebraically and graphically.
8.
x 2  7 x  12
24.
x4  4 x2
46.
x2
1
x4
18.
 x  2   x 2  x  1  0
38.
x 3
0
x 1
50.
5
3

x  3 x 1