Review: 3. What is an asymptote? 4. What is an end behavior model?

Slides:

Advertisements

Similar presentations

Chapter 4: Polynomial & Rational Functions 4.4: Rational Functions

Advertisements

W-up Get out note paper Find 12.3 notes

Section 3.2 Notes Writing the equation of a function given the transformations to a parent function.

2.2 Limits Involving Infinity. Graphically What is happening in the graph below?

Rational Functions. 5 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros.

2.6 Limits at Infinity. |x|x As x approaches infinity f(x)  ? as x gets larger and larger f(x)  ? as x gets larger and larger in the negative direction.

LIMITS INVOLVING INFINITY Mrs. Erickson Limits Involving Infinity Definition: y = b is a horizontal asymptote if either lim f(x) = b or lim f(x) = b.

Lesson 2.2 Limits Involving Infinity  Finite Limits as x->∞  Sandwich Theorem Revisited  Infinite limits as x -> a  End Behavior Models  “Seeing”

10.2: Infinite Limits. Infinite Limits When the limit of f(x) does not exist and f(x) goes to positive infinity or negative infinity, then we can call.

10/13/2015Mrs. Liu's PreCalc Day191 CP Calculus Block 19 HW-Review p88, 84;90;96;108;114; 120;126;

5.1 Polynomial Functions Degree of a Polynomial: Largest Power of X that appears. The zero polynomial function f(x) = 0 is not assigned a degree.

9.3 Rational Functions and Their Graphs Rational Function – A function that is written as, where P(x) and Q(x) are polynomial functions. The domain of.

1.5 Infinite Limits IB/AP Calculus I Ms. Hernandez Modified by Dr. Finney.

1.5 Infinite Limits and 3.5 Limits at Infinity AP Calculus I Ms. Hernandez (print in grayscale or black/white)

Asymptotes Objective: -Be able to find vertical and horizontal asymptotes.

Class Work Find the real zeros by factoring. P(x) = x4 – 2x3 – 8x + 16

1.5 Infinite Limits Objectives: -Students will determine infinite limits from the left and from the right -Students will find and sketch the vertical asymptotes.

What is the symmetry? f(x)= x 3 –x.

Properties of Rational Functions 1. Learning Objectives 2 1. Find the domain of a rational function 2. Find the vertical asymptotes of a rational function.

Limits Involving Infinity Section 2.2. ∞ Infinity Doesn’t represent a real number Describes the behavior of a function when the values in its domain or.

8-2: Exponential Decay Objective Ca Standard 12: Students know the laws of fractional exponents, understand exponential functions and use these functions.

3.4 Properties of Logarithmic Functions

FUNCTION TRANSLATIONS ADV151 TRANSLATION: a slide to a new horizontal or vertical position (or both) on a graph. f(x) = x f(x) = (x – h) Parent function.

Lesson 3.2. Parent Function General Formula We can write all parent functions in the following form:

Pg. 223/224/234 Homework Pg. 235 #3 – 15 odd Pg. 236#65 #31 y = 3; x = -2 #33y = 2; x = 3 #35 y = 1; x = -4#37f(x) → 0 #39 g(x) → 4 #41 D:(-∞, 1)U(1, ∞);

Section 2.2a. Limits Involving Infinity We can say “the limit of f as x approaches infinity,” meaning the limit of f as x moves increasingly far to the.

1.5 Infinite Limits. Find the limit as x approaches 2 from the left and right.

Section Shifting Functions & Periodicity. OBJECTIVES SWBAT: SWBAT explain the effect of shifts for piecewise functions. Shift functions (including.

2.1 Domain and Range of Functions. Vocabulary ● Relation – A relation is a general term for any set of ordered pairs. ● Function – A function is a special.

The base e P 667. Essential Question How is the graph of g(x) = ae x – h + k related to the graph of f(x) = e x.

Rational Functions. 6 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros 6)Slant Asymptotes.

Infinite Limits Unit IB Day 5. Do Now For which values of x is f(x) = (x – 3)/(x 2 – 9) undefined? Are these removable or nonremovable discontinuities?

Infinite Limits 1.5. An infinite limit is a limit in which f(x) increases or decreases without bound as x approaches c. Be careful…the limit does NOT.

REVIEW. A. All real numbers B. All real numbers, x ≠ -5 and x ≠ -2 C. All real numbers, x ≠ 2 D. All real numbers, x ≠ 5 and x ≠ 2.

A rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = . Its graph is a.

Do Now from 1.2b Find all values of x algebraically for which the given algebraic expression is not defined. Support your answer graphically. and.

Rational Functions A rational function has the form

Ch. 2 – Limits and Continuity

Limits Involving Infinity

4.4 Rational Functions A Rational Function is a function whose rule is the quotient of two polynomials. i.e. f(x) = 1

W-up 3-27 When is the function increasing?

Logarithmic Functions

2.2 Limits involving Infinity Day 1

Rational functions are quotients of polynomial functions.

Ch. 2 – Limits and Continuity

Graphing Square Root Functions

26 – Limits and Continuity II – Day 2 No Calculator

MATH 1310 Session 8.

Graphing Polynomial Functions

2.2 Limits Involving Infinity

2.7 Graphs of Rational Functions

Exponential Functions and Their Graphs

Exponential Functions and Their Graphs

True or False: Suppose the graph of f is given

Limits involving infinity

4.2 Exponential Functions and Their Graphs

True or False: {image} is one-to-one function.

Limits and Continuity Chapter 2:.

3.1 Exponential Functions and Their Graphs

A rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = . Its graph is a.

Exponential Functions and Their Graphs

Exponential Functions and Their Graphs

2.7 Graphs of Rational Functions

Pre-Calculus Go over homework End behavior of a graph

27 – Graphing Rational Functions No Calculator

Exponential Functions and Their Graphs

Product and Composition of Limits

Exponential Functions and Their Graphs

Ch. 11 Vocabulary 7.) Rational function 8.) Asymptote.

Warm up honors algebra 2 3/1/19

Presentation transcript:

Review: 3. What is an asymptote? 4. What is an end behavior model?

Answers: ft/s Does not exist / Possible Answer [-4.7,4.7] [-15,15] a,d,e,f,g,h,i: True b,c: False 4 7. y = x - 1, x≠1 50a. 4 b. -21 c. -12 d. -7/3 51b. c. No 54b. c. Yes 55b(-2π,0) ∪ (0,2π) c. 2π d. -2π a. y = gt 2 g = 1.25 b. 5m/s c. 10 m/s

Vocabulary: Horizontal Asymptote: Occurs if or for any line y = b Vertical Asymptote: Occurs if or for any line x = a End Behavior Model: Take two different functions that are being divided and use limits to estimate what value they will approach.

Properties of Limits as x ±∞ If L, M, and k are real numbers and

Ex 1: Find the horizontal asymptotes for

Ex 2: Find

Ex 3: Find c. Does the limit as x approaches 2 exist? Why or why not?

Ex 4: Find the end behavior for: a) f(x) = 4x 5 - 5x 3 + 9x - 11 b) g(x) = -4x 2 + 8x 3 - x 4 c) h (x) = 6x x x -1

Ex 5: Find the end behavior for: a) f(x) = 5x x + 25b) g(x) = 4x 3 - 8x x 3 + 9xx 2 + 5

Ex 6: Find the left and right behavior for a) f(x) = e x - 2xb) g(x) = x 2 + sin x

Assignment: p76 #3-36(x3), 43, 46, 54-55, 59, 65, 68