Graph Rational Functions

Slides:

Advertisements

Similar presentations

9.2 Graphing Simple Rational Functions

Advertisements

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.

3.4 Rational Functions I. A rational function is a function of the form Where p and q are polynomial functions and q is not the zero polynomial. The domain.

Rational Expressions, Vertical Asymptotes, and Holes.

G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY Let p (x ) and q (x ) be polynomials with no common factors other than 1.The.

Warm-Up: FACTOR 1.x 2 – x x x 2 – x – 2 5.x 2 – 5x – x 2 – 19x – 5 7.3x x - 8.

EXAMPLE 1 Graph a rational function of the form y = a x Graph the function y =. Compare the graph with the graph of y =. 1 x 6 x SOLUTION STEP 1 Draw the.

Section 5.2 – Properties of Rational Functions

Objectives: Find the domain of a Rational Function Determine the Vertical Asymptotes of a Rational Function Determine the Horizontal or Oblique Asymptotes.

EXAMPLE 1 Graph a rational function (m < n) Graph y =. State the domain and range. 6 x SOLUTION The degree of the numerator, 0, is less than the.

ACT Class Openers:

3.6 Warm Up Find the initial point, state the domain & range, and compare to the parent function f(x) = √x. y = 3√x – 1 y = -1/2√x y = - √(x-1) + 2.

3.6 Graph Rational Functions Part II. Remember Rational functions have asymptotes To find the vertical asymptote, set the denominator = 0 and solve for.

Objectives: Students will be able to… Graph simple rational functions Determine domain and range of rational functions.

EXAMPLE 2 Graph an exponential function Graph the function y = 2 x. Identify its domain and range. SOLUTION STEP 1 Make a table by choosing a few values.

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.

2.6 Rational Functions and Asymptotes 2.7 Graphs of Rational Functions Rational function – a fraction where the numerator and denominator are polynomials.

Section 5.2 Properties of Rational Functions

Rational Functions Intro - Chapter 4.4.  Let x = ___ to find y – intercepts A rational function is the _______ of two polynomials RATIO Graphs of Rational.

Rational Functions Essential Questions

Rational Functions Objective: Finding the domain of a rational function and finding asymptotes.

Do Now: State the domain of the function.. Academy Algebra II 7.1, 7.2: Graph Exponential Growth and Decay Functions HW: p.482 (6, 10, even), p.489.

Objectives: 1. Be able to identify the parent function for a rational. 2.Be able list the characteristic of a rational function. 3.Be able to graph rational.

EXAMPLE 1 Compare graph of y = with graph of y = a x 1 x 1 3x3x b. The graph of y = is a vertical shrink of the graph of. x y = 1 = y 1 x a. The graph.

2.6. A rational function is of the form f(x) = where N(x) and D(x) are polynomials and D(x) is NOT the zero polynomial. The domain of the rational function.

Warm up:. Notes 7.2: Graphing Rational Functions.

Notes Over 9.2 Graphing a Rational Function The graph of a has the following characteristics. Horizontal asymptotes: center: Then plot 2 points to the.

Graphing Rational Expressions. Find the domain: Graph it:

Check It Out! Example 2 Identify the asymptotes, domain, and range of the function g(x) = – 5. Vertical asymptote: x = 3 Domain: {x|x ≠ 3} Horizontal asymptote:

Lesson 21 Finding holes and asymptotes Lesson 21 February 21, 2013.

9.2 Graphing Simple Rational Functions Obj: to graph a hyperbola Do Now: What value(s) would make this function undefined? x = -4 and y = -7.

8.2 The Reciprocal Function Family Honors. The Reciprocal Functions The Reciprocal function f(x) = x ≠0 D: {x|x ≠ 0} R: {y|y ≠ 0} Va: x = 0 Ha: y = 0.

Characteristics of Rational Functions

Section 2.6 Rational Functions Part 2

Rational Functions.

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

GRAPHING RATIONAL FUNCTIONS

Graphing Rational Functions

8.1/8.2- Graphing Rational Functions

8.2 Rational Functions and Their Graphs

Section 6.2 – Graphs of Exponential Functions

Rational functions are quotients of polynomial functions.

exponential functions

Graph Simple Rational Functions

Warm-Up: FACTOR x2 – 36 5x x + 7 x2 – x – 2 x2 – 5x – 14

Quote of the Day What is now proved was once only imagined. -William Blake.

11-6B Graph Inverse variation (Simple Rational Functions)

8.2 Graph Simple Rational Functions

Rational Functions and Asymptotes

Graphing a Simple Rational Function

Graphing Exponential Functions

Graphing Rational Functions

Notes Over 9.3 Graphing a Rational Function (m < n)

Factor completely and simplify. State the domain.

Attributes and Transformations of Reciprocal Functions

Simplifying rational expressions

2.6 Section 2.6.

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.

Simplifying rational expressions

Rational Functions Essential Questions

3.4 Rational Functions I.

Grab a calculator and graph the following equations:

Graphing Rational Expressions

Domain, Range, Vertical Asymptotes and Horizontal Asymptotes

Section 8.4 – Graphing Rational Functions

Solving and Graphing Rational Functions

8.2 Graph Simple Rational Functions

4.3 Rational Functions I.

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

Presentation transcript:

Graph Rational Functions 3.6 Graph Rational Functions Vocabulary A rational function has a rule given by a fraction whose numerator and denominator are polynomials and whose denominator is not 0. Rational Function A line that a graph approaches more and more closely. Asymptote

Graph Rational Functions 3.6 Graph Rational Functions Example 1 The graph of y = 2/x is a vertical ______________________________ of the graph of y = 1/x . stretch

Graph Rational Functions 3.6 Graph Rational Functions Example 2 The graph of y = -1/4x is a vertical __________ with a reflection in the ________ of the graph of y = 1/x . shrink x-axis Note that the graph could also be viewed as being reflected in the ________ . y-axis

Graph Rational Functions 3.6 Graph Rational Functions Checkpoint. Complete the following exercise. Compare the graph of with the graph of The graph is a vertical stretch With a reflection in the x-axis of the graph of

Graph Rational Functions 3.6 Graph Rational Functions The graph has the following characteristics: If |a| > 1, the graph is a vertical ________ of the graph of stretch If 0 < |a| < 1, the graph is a vertical ________ of the graph of shrink If a < 0, the graph is a reflection in the ________ of the graph of x-axis The horizontal asymptote is y = __. k The vertical asymptote is x = __. h The domain of the function is all real numbers except x = __. h The range of the function is all real numbers except y = __. k

Graph Rational Functions 3.6 Graph Rational Functions Example 5 Step 1 Identify the asymptote of the graph. 15 9 3 The vertical asymptote is x = __. 3 The horizontal asymptote is y = __. 4 Step 2 Plot several points on each side of the ________ asymptote. -9 -3 3 9 15 vertical Step 3 Graph two branches that pass through the plotted points and approach the ____________. asymptote

Graph Rational Functions 3.6 Graph Rational Functions Checkpoint. Complete the following exercise. 9 3 -3 -9 -3 3 9

Graph Rational Functions 3.6 Graph Rational Functions