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

qq

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]