Upload
others
View
22
Download
1
Embed Size (px)
Citation preview
3.1 Power Functions & Polynomial Functions
A power function is a function that can be represented in the form
𝑓(𝑥) = 𝑥𝑝, where the base is a variable and the exponent, p, is a number.
The Effect of the Power P
Positive integer power
(Even) 𝑦 = 𝑥2 𝑣𝑠 𝑥4 (Odd) 𝑦 = 𝑥3 𝑣𝑠 𝑥5
Negative odd integer powers Negative even integer powers
𝑦 = 𝑥0 = 1
𝑦 = 𝑥1 = 𝑥
The same characteristic U-shaped.
Symmetric about the y- axis.
𝑥4 is flatter near the origin and steeper away from
the origin than 𝑥2.
The same characteristic “chair”-shaped.
Symmetric about the origin.
𝑥5 is flatter near the origin and steeper away from
the origin than 𝑥3.
Polynomial Functions and Their Graphs
Ex 1: Consider the function 573 45102 xxxxxP and answer the following:
Write P in standard form: ___________________________ The degree of this polynomial: ______________ The leading term: ________________________ The leading coefficient: ___________________ Constant: ______________________________
an: ______ a7: ______ a6: ______ a5: ______ a4: ______
a3: ______ a2: ______ a1: ______ a0: ______
Ex 2: Which of the following are polynomial functions: If they are polynomial, state the degree, an and a0
Polynomial: Degree an a0 10023 xxxP _________ _______ ________ _______
112 xxxQ _________ _______ ________ _______
385xxR _________ _______ ________ _______
5xS _________ _______ ________ _______
xxxT 105.2 _________ _______ ________ _______
64 xxU _________ _______ ________ _______
152 xxV x
_________ _______ ________ _______
xxW 1 _________ _______ ________ _______
Graph of polynomial: Polynomial functions must be smooth and continuous functions.
By smooth, the graph contains no sharp corners or cusps.
By continuous, the graph has no gaps or holes.
Turning Points Theorem: If f is a polynomial function of degree n, then f has at most (n-1) turning points.
If the graph of a polynomial function f has (n-1) turning points, the degree of f is at least n.
The Long-Run Behavior (End Behavior) of Polynomial The behavior of the graph of a function as the input takes on large negative values ( x ) and large positive values
( x ) as is referred to as the long run behavior of the function
Ex. Describe the end behavior of the function f(x) = 3x3 – 2x4 – 3x + 16 by completing the following statements:
a) As x → ∞, f(x) → ____________? b) As x → -∞, f(x) → ___________ ?
Short Run Behavior Characteristics of the graph such as vertical (y-intercept) and horizontal intercepts (x-intercepts) and the places the
graph changes direction are part of the short run behavior of the polynomial.
Ex. Find the vertical and horizontal intercepts of each function.
32. 3 1 4 ( 5)f x x x x 34. )34(43 nnnk
3.2 Quadratic Functions Forms of Quadratic Functions
The standard form of a quadratic function is cbxaxxf 2)(
The transformation form/ vertex form of a quadratic function is khxaxf 2)()(
The vertex of the quadratic function is located at (h, k), where h and k are the numbers in the transformation
form of the function. Because the vertex appears in the transformation form, it is often called the vertex form.
Ex1. Use the given graph of 𝑓 to find the following:
Solve for x:
Ex2a. 016
11
2
2
xx
Ex2b. 113 xx
= 𝑎(𝑥 − ℎ)2 + 𝑘
Vertex: ______________________
Classify vertex: lowest point or highest point? ________
Graph has: minimum y-value or maximum y-value, which is _________
Does the graph open upward or downward? ___________
Axis of symmetry: ____________
Find the vertex, minimum or maximum, vertical and horizontal intercepts, axis of symmetry, rewrite the quadratic
function into vertex form, and graph f(x).
Ex3. Let 432 xxxf .
Determine if the graph of 𝑓 opens upward or downward? _______
The Vertex is:________________
The minimum or maximum value is: _________________
The Axis of symmetry: ________________
Vertical intercept (as a point): _________________________
Horizontal intercept(s) (as point(s)): _____________________________
Vertex form: _________________________________________
Ex4. Let xxxf 182 2 .
Determine if the graph of 𝑓 opens upward or downward? _______
The Vertex is:________________
The minimum or maximum value is: _________________
The Axis of symmetry: ________________
Vertical intercept (as a point): _________________________
Horizontal intercept(s) (as point(s)): _____________________________
Vertex form: _________________________________________
Write an equation for a quadratic with the given features
21. x-intercepts (2, 0) and (5, 0), and y intercept (0, 6) 25. Vertex at (-3, 2), and passing through (3, -2)
27. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by
24.9 229 234h t t t .
a. From what height was the rocket launched?
b. How high above sea level does the rocket reach its peak?
c. Assuming the rocket will splash down in the ocean, at what time does splashdown occur?
3.3 Graphs of Polynomial Functions
Ex1. Describe the end behavior of the function f(x) = 5x4 – x
5 – 3x + 2x
2 -15 by completing the following statements:
a) Graph the End behavior: __________ b)As x → ∞, f(x) → ________? c) As x → -∞, f(x) → ________ ?
Writing Equations using Intercepts Factored Form of Polynomials
If a polynomial has horizontal intercepts at nxxxx ,,, 21 , then the polynomial can be written in the
factored form np
n
ppxxxxxxaxf )()()()( 21
21
where the powers pi on each factor can be determined by the behavior of the graph at the corresponding
intercept, and the stretch factor a can be determined given a value of the function other than the horizontal
intercept.
Ex. Write an equation for a polynomial the given features. 31. Degree 3. Zeros at x = -2, x = 1, and x = 3. Vertical intercept at (0, -4)
Ex2: 12)( xxxP
Multiplicity (Repeated, or multiple zero of P)
If mcx is a factor of a polynomial P and 1
mcx is not a factor of P, then c is called a zero of multiplicity m of P.
Note: If m = even, the graph of P touches the x-axis at c. If m = odd, the graph of P crosses the x-axis at c. 33. Degree 5. Roots of multiplicity 2 at x = 3 and x = 1, and a root of multiplicity 1 at x = -3. Vertical intercept at (0, 9)
35. Degree 5. Double zero at x = 1, and triple zero at x = 3. Passes through the point (2, 15)
Solve each inequality.
19. 2
3 2 0x x 21. 1 2 3 0x x x
m = 1, straight cross m > 1, flattened appearance.
Ex3. Answer the following of the polynomial f .
𝑓(𝑥) = (𝑥 − 4)(3𝑥 − 2)2 Degree of f:____________
an= ______________
a0= ______________
Graph the End behavior: ________________
𝑓(𝑥) = −2(𝑥 − 1)4(3 − 2𝑥)2 Ex4. Answer the following of the polynomial f .
Degree of f:____________
an= ______________
a0= ______________
Graph the End behavior: ________________
Zeros:
Multiplicity:
Cross or Touch:
Zeros:
Multiplicity:
Cross or Touch:
Ex5. Answer the following of the polynomial f, and then graph the function.
212)(32
xxxxf
Degree of f:____________ an= ______________ Graph the End behavior: ________________ Test points: y-intercept (as a point): __________________ Ex. Write a formula for each polynomial function graphed. 41.
47.
Zeros:
Multiplicity:
Cross or Touch:
3.4 Rational Functions
Dividing Polynomials
Long Division of Polynomials Divide 842 by 15.
Synthetic Division Synthetic Division can be used to divide (x-k) into a polynomial.
For example,
𝑎𝑥2 + 𝑏𝑥 + 𝑐
𝑥 − 𝑘
Ex. 5𝑥4−2𝑥2+39𝑥−1
𝑥+2 Ex.
𝑥4−81
𝑥+3
Check:
Is 15 ∗ 56 + 2 842=? ?
Note:
If the dividend is missing a
term, we will replace that
term with 0 to allow the terms
to line up while we do the
division process.
𝑄(𝑥) +𝑅(𝑥)
𝐷(𝑥)
Answer should be in the form
of:
Rational Functions
𝑟(𝑥) =𝑃(𝑥)
𝑄(𝑥)=
𝑎𝑛𝑥𝑛+𝑎𝑛−1𝑥𝑛−1+⋯+𝑎1𝑥+𝑎0
𝑏𝑚𝑥𝑚+𝑏𝑚−1𝑥𝑚−1+⋯+𝑏1𝑥+𝑏0, where
𝑃(𝑥), and 𝑄(𝑥) are polynomial functions and 𝑄(𝑥) ≠ 0.
Ex.𝑟(𝑥) =6𝑥2+2𝑥+3
𝑥+2 Domain: _______________________
Ex. 𝑟(𝑥) =3𝑥3+𝑥−1
𝑥2−1 Domain: _______________________
Vertical Asymptotes (Short run behavior)
Locating Vertical Asymptotes
A rational function 𝑟(𝑥) =𝑃(𝑥)
𝑄(𝑥), in lowest terms, will have a vertical asymptotes 𝑥 = 𝑎 if a is a real zero of the
denominator Q.
Holes of Rational Functions A hole might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. In
this case, factor the numerator and denominator and simplify; if the simplified expression still has a zero in the
denominator at the original input the original function has a vertical asymptote at the input, otherwise it has a hole.
Ex.
Domain: __________________________
Note: The domain of 𝑟(𝑥) is the set of all
real numbers excepts those for which the
denominator Q is 0.
𝑟(𝑥) =(𝑥 − 2)(𝑥 + 1)
𝑥 + 1
Ex.𝑟(𝑥) =−2𝑥2+1
2𝑥3+4𝑥2
V.A: _______________________
Transformations of
Transformations of
Graph of Graph of Graph of
Graph of Graph of Graph of
𝑦 =1
𝑥
𝑦 = −1
𝑥 𝑦 =
1
𝑥 − 2 𝑦 = −
1
𝑥 + 3
𝑦 =1
𝑥2
𝑦 = −1
𝑥2 y =
1
(𝑥 − 2)2 y = −
1
(𝑥 + 3)2
Finding Horizontal and Oblique (Slant) Asymptotes of a Rational Function
No Horizontal and Oblique (Slant) Asymptotes(𝒏 − 𝒎 ≥ 𝟐):
If 𝑛 − 𝑚 ≥ 2, 𝑟 has no Horizontal and Oblique (Slant)
Asymptotes. In this case, for |𝑥| unbounded, 𝑥 → ±∞,
the graph of r will behave like the graph of the quotient.
Ex. 3 2
1
xr x
x
Oblique (Slant) Asymptotes (𝒏 − 𝒎 = 𝟏):
If 𝑛 − 𝑚 = 1, the quotient obtained is of the form
𝑎𝑥 + 𝑏, and the line 𝑦 = 𝑎𝑥 + 𝑏 is the oblique
asymptote.
Ex. 2 2 4
1
x xr x
x
3 2
1
xr x
x
2 1Q x x x
2 2 4
1
x xr x
x
Horizontal Asymptote
If 𝑛 = 𝑚, then the graph will have the horizontal asymptote 𝑦 =𝑎𝑛
𝑏𝑚.
Ex. 2
2
6 12
3 5 2
x xr x
x x
If 𝑛 < 𝑚, then the graph will have the horizontal asymptote 𝑦 = 0.
Ex. 2
3 2
2 1
2 4
xr x
x x
2
2
6 12
3 5 2
x xr x
x x
2
3 2
2 1
2 4
xr x
x x
Ex. Of the rational function, find all the x- and y- intercepts; the verticalasymptotes; the behavior near by the vertical asymptotes; the horizontal or oblique asymptotes; the intersection between the function and the horizontal or oblique asymptotes ,if any; and obtain the graph of r.
56
42
2
xx
xxxr
Factor:
x-intercepts:
y-intercepts:
V.A:
Behavior nearby the V.A:
H.A or O.A:
Find the intersection between the graph and H.A or O.A:
Note:
The graph of a function will never intersect a vertical asymptote.
The graph of a function may intersect a horizontal asymptote or an oblique asymptote.
If 𝑛 − 𝑚 ≥ 2, 𝑟 has no Horizontal and Oblique (Slant) Asymptotes.
If 𝑛 − 𝑚 = 1, r has Oblique Asymptote, 𝑦 = 𝑎𝑥 + 𝑏.
If 𝑛 = 𝑚, r has Horizontal Asymptote .
If 𝑛 < 𝑚, r has Horizontal Asymptote 𝑦 = 0.
𝑦 =𝑎𝑛
𝑏𝑚
Ex:
4
122
2
x
xxxr
Factor:
x-intercepts:
y-intercepts:
V.A:
Behavior nearby the V.A:
H.A or O.A:
Find the intersection between the graph and H.A or O.A:
Writing Rational Functions from Intercepts and Asymptotes
If a rational function has horizontal intercepts at nxxxx ,,, 21 , and vertical asymptotes at mvvvx ,,, 21
then the function can be written in the form
n
n
q
m
p
n
pp
vxvxvx
xxxxxxaxf
)()()(
)()()()(
21
21
21
21
where the powers pi or qi on each factor can be determined by the behavior of the graph at the corresponding
intercept or asymptote, and the stretch factor a can be determined given a value of the function other than the
horizontal intercept, or by the horizontal asymptote if it is nonzero.
Write an equation for the function graphed.
32. 34.
3.5 Inverses and Radical Functions In this section, we will explore the inverses of polynomial and rational functions, and in particular the radical functions
that arise in the process.
When we try to find the inverse of polynomial functions, we have a slight difficulty: Because most polynomial functions
are not one- to- one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these
functions so that they become one- to- one.
For Example:
Is the function 𝑓(𝑥) = 𝑥2 one- to-one? Is the function ℎ(𝑥) = 𝑥2, 𝑥 ≥ 0, one- to-one?
Exercises: For each function, find a domain on which the function is one-to-one and non-decreasing, then find an inverse
of the function on this domain.
2. 2
2f x x 6. 324 xxf
Exercises: Find the inverse of each function.
8. 6 8 5f x x 10. 33f x x
14. 2
7
xf x
x
16.
5 1
2 5
xf x
x
21. A drainage canal has a cross-section in the shape of a
parabola. Suppose that the canal is 10 feet deep and 20 feet
wide at the top. If the water depth in the ditch is 5 feet, how
wide is the surface of the water in the ditch? [UW]