name: date: exploration 4-1 introduction to polynomial & rational …€¦ · & rational...

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Exploration 4-1 Introduction to Polynomial & Rational Functions

Objective: Discover properties of polynomial and rational functions and their graphs.

𝑓(𝑥) = 𝑥2 − 2𝑥 − 8 𝑔(𝑥) =𝑥3−5𝑥2−2𝑥+25

𝑥−3

1. Function 𝑓 appears to have x-intercepts at -2 and

4. Confirm by direct substitution that 𝑓(−2) and 𝑓(4) both equal zero. Then explain why -2 and 4 are also called zeros.

2. The x-value -2 appears to be a zero of the function 𝑔. Confirm or refute this observation numerically.

3. Function 𝑔 has a vertical asymptote at 𝑥 = 3. Try

to find 𝑔(3). Then explain why 𝑔(3) is undefined. Explain the behavior of 𝑔(𝑥) when x is close to 3 by

finding 𝑔(3.001) 𝑎𝑛𝑑 𝑔(2.999).

4. Plot the graphs of the functions 𝑓 𝑎𝑛𝑑 𝑔 on the same screen. What do you notice about the two graphs when x is far from 3? When x is close to 3? How could you describe the relationship of the graph of function 𝑓 to the graph of function 𝑔?

5. Let function ℎ be the vertical translation of

function 𝑓 by 10 units. That is, ℎ(𝑥) = 𝑓(𝑥) + 10. Show by graphing that no zeros of function ℎ are real numbers.

6. Explain why a quadratic function can have no more than two real numbers as zeros.

7. As a challenge, show that the complex number,

1 + 𝑖 (𝑤ℎ𝑒𝑟𝑒 𝑖 = √−1) is a zero of ℎ(𝑥). Explain why

a function can have a zero that is not an x-intercept.

8. What did you learn as a result of doing this problem set?

Pre-Calculus Chapter 4 Notes

Section 4.2 Notes

Expanding Two Binomials

Distributive Property (________) Box Method

(3𝑥 + 5)(2𝑥 − 11) (2𝑥 − 4)2

Quadratic Formula

𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 Example: Solve for the zeros of the function h(x).

𝒙 = ℎ(𝑥) = 𝑥2 − 4𝑥 + 13

Imaginary/Complex Numbers

√−1 = ______ 𝑡ℎ𝑎𝑡 𝑚𝑒𝑎𝑛𝑠 − 1 = ________

Complex number: 𝒂 + 𝒃𝒊

The complex numbers ______________ and ________________ are called ___________________________

__________________________________ of each other. Example: 2 + 3𝑖 𝑎𝑛𝑑 _______________.

Example: Solve the following quadratic equation,

using the quadratic formula.

𝑥2 + 6𝑥 + 30 = 17

Example: Multiply and simplify.

(7 + 4𝑖)(9 + 10𝑖)


Quadratic Forms

Standard Form Vertex Form

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘

Vertex:

Example: Example:

* Vertex form helps us see

____________________________!

Vertex form to Standard form

𝑦 = 3(𝑥 − 5)2 − 43

Standard form to Vertex form

(Completing the square)

𝑓(𝑥) = 𝑥2 + 6𝑥 + 11

Set the equation equal to zero, and move the

“c” term.

Make sure a=1

Take half of b and square that answer

Add the result to both sides of the equation

Factor the one side of the equation, and move

everything to one side.

Example: Solve the following quadratic equation by completing the square.

0 = 𝑥2 − 12𝑥 + 27


Section 4.3 Notes

Degree- ___________________________________________________________________

Leading Coefficient- _________________________________________________________________

__________________________________________________________________________________

Examples:

The end behavior is _______, therefore the leading coefficient is ______________________.

y=____________________ y=_________________________

y=_________________________

Examples:

The end behavior is _________, therefore the leading coefficient is ____________________.

y=____________________ y=_________________________

y=_________________________


Types of Real Zeros

Match the zeros to their corresponding graph.

One set of zeros will not be used.

A. 𝑥 = 5, −10 Graph: _____

B. 𝑥 = 2, 2, −10 Graph: _____

C. 𝑥 = 2, 2, −10, −4𝑖, 4𝑖 Graph: _____

D. 𝑥 = 5, −10, −2𝑖, 2𝑖 Graph: _____

Dividing Polynomials

Writing in Mixed Number Form

Synthetic Division/Substitution Example: Divide 𝑓(𝑥) = 𝑥3 − 9𝑥2 − 𝑥 + 105 by 𝑥 − 6

A B

C


Long Division Example: Divide 𝑓(𝑥) = 𝑥3 − 9𝑥2 − 𝑥 + 105 by 𝑥 − 6

Extreme Point(________________ Point): _____________________________

_______________________________________________________________

Calculator Steps:

Sums/Products of zeros of Cubic Functions

If 𝑝(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 has the zeros 𝑧1, 𝑧2, 𝑎𝑛𝑑 𝑧3 then,

Sum:

Product:

Sum of Pairwise Products:


Section 4.4 Notes

Inflection Point: ______________________________________________________________________

Example : 𝑓(𝑥) = −2𝑥3 + 15𝑥2 − 19𝑥 + 44

X-Coordinate of a Cubic

Function’s Inflection

Point:


Section 4.5 Notes

Rational Functions

Discontinuities

Non-Vertical Asymptotes

Improper fractions can be written in __________________________________ by dividing the

polynomials. This can be done using synthetic or long division.

Mixed Number Form: 𝑓(𝑥) =


Example: Find the non-vertical asymptotes of the function given below.

𝑓(𝑥) =𝑥2 + 4𝑥 − 5

𝑥2 − 8𝑥 + 7

Horizontal Asymptotes

Slant Asymptotes

Example: Use polynomial division to find the non-vertical asymptote of the rational function given below.

𝑓(𝑥) =𝑥2−16

𝑥−5

X & Y Intercepts

X-Intercepts: _________________________________ Y-Intercepts: _________________________________


Sketching Rational Functions

Example: 𝒈(𝒙) =−𝟐𝒙𝟐+𝟑𝒙+𝟐𝟎

𝒙𝟐−𝟏𝟔

Removeable Discontinuity x-intercepts

y-intercepts

Vertical Asymptotes

Mixed Number Form Non-Vertical Asymptotes

Example: 𝒈(𝒙) =𝒙𝟐−𝟒

𝒙𝟐+𝟒


y-intercepts

Vertical Asymptotes



Example: 𝒈(𝒙) =𝒙𝟐−𝟔𝒙+𝟖

𝒙−𝟏


y-intercepts

Vertical Asymptotes



Section 4.6 Notes

Operations with Rational Functions

Example: Multiply and simplify. Example: Divide and simplify.

Example: Add the following. Example: Subtract the following.


Solving Rational Equations

Example: If 𝑓(𝑥) =𝑥2−5𝑥+10

𝑥−3, find when 𝑓(𝑥) = 6.

Graphically Algebraically

Domain Restrictions:

Example: If 𝑓(𝑥) =𝑥2−5𝑥+10

𝑥−3, find when 𝑓(𝑥) = 4.



Example: Find where 𝑓(𝑥) =𝑥

𝑥−2+

2

𝑥+3 and 𝑔(𝑥) =

10

𝑥2+𝑥−6 intersect.




Solving Fractional Equations

Example:

𝑥 + 3

2𝑥 − 3=

18𝑥

4𝑥2 − 9

Example:

𝑥

𝑥 + 4+

4

𝑥 − 4=

𝑥2 + 16

𝑥2 − 16

name: date: exploration 4-1 introduction to polynomial & rational …€¦ · & rational...

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