Introduction to Rational Equations

2 Types of Functions•Continuous•Discontinuous

Continuous

Continuous

• Keeps going• No breaks in graph• Smooth

Discontinuous

Discontinuous

• Stops• Graph has breaks or holes

Examples

• Continuous Graphs → Polynomials• Discontinuous Graphs → Rational

Equations

Examples

Your Turn: Be Prepared to Share!!!• Complete problems 1 – 6 on the Introduction

to Rational Equations handout• Remember, you need to:– Classify the graph as either continuous or

discontinuous– Classify the graph as either a polynomial or a

rational equation– Justify your reasoning!!!

Sharing Activity1. I will gently throw the ball to a student.2. That student answers the first question.3. Then the student will gently throw the ball to

another student.4. That student answers the next question.5. Repeat until we’ve answered all the

questions.

Polynomial

• Monomial• Binomial• Trinomial

•Polygamy•Polytheism•Polydactyl•Polyglot

Polynomials, cont.• A polynomial is algebraic expression that can

be written in the formanxn + an-1xn-1 + … + a2x2 + a1x1 + a0

• An equation or an expression with a single variable raised to (usually many) powers

• All exponents are whole numbers• an ≠ 0 (Leading Coefficient ≠ 0)

Polynomial Examples• Generally a long list of variables• f(x) = x4 – 4x3 + 2x2 – 3x + 11• f(x) = x11 + 7x5 – 4x3 + x – 12

• But we can also have a short list of variables• f(x) = x5 + x• f(x) = x2 – 1

• Or even no variables at all!• f(x) = 10• f(x) = ½

Rational Equation

PolynomialPolynomial

Rational

Rational Equations, cont.

• Rational equations are fractions in which both the numerator and the denominator are polynomials

• We don’t need variables in the numerator, but we must have them in the denominator!!!

Rational Examples

423

2

xx

xx)x(f

42

xx

)x(f

1

13

xx

)x(f4

2x)x(f

Polynomials vs. Rational Equations

1. f(x) = x8 – 7x2 + 4 2. f(x) = 11

3. 4.

5.

162

3

xx

)x(f xxx

)x(f 23

23

xx)x(f

412

Your Turn: Be Prepared to Share!!!

• Complete problems 7 – 12 on the Introduction to Rational Equations handout.

• Remember, you need to:– Classify the equation as either a polynomial or a

rational equation.– Justify your reasoning

Compare – Contrast – Summarize Graphic Organizer

Continuity → Continuous or Discontinuous

How Alike?

How Different?

Polynomials

With Regard to Graphs

Rational Equations

How Different?

Polynomials

With Regard to Equations

Rational Equations

How Different?

Polynomials

With Regard to Continuity

Rational Equations

Summarize:

Discontinuous Graphs

Discontinuities

Rational Graphs

Discontinuities

• Discontinuity – a point or a line where the graph of an equation has as a hole, a jump, a break, or a gap

• Affect the shape, domain and range of an equation

*Discontinuities, cont.

• Three major types of discontinuities:

• Vertical Asymptotes

• Horizontal Asymptotes

• Holes

Asymptotes

Point (Removable) Discontinuity

Type of Discontinuities – Asymptotes• Lines that the graph

approaches but (almost) never crosses

• Represented by a dashed line

• Not part of the equation

• We don’t draw them if they happen on either the x-axis or the y-axis

Vertical Asymptotes (1st Column)

• Occur when the denominator equals zero• Can never be crossed• Always in the form x = • Abbreviated VA

Vertical Asymptotes, cont.Hand Drawn Calculator Drawn

The calculator doesn’t draw the asymptotes!!!!

Horizontal Asymptotes (2nd Column)• Occur when the degree of the denominator is ≥

the degree of the numerator

• Ex. • Can be crossed when |x| is very large• Describes the end behavior of a rational equation• Always in the form y = • Abbreviated HA

3xx

y

Horizontal Asymptotes, cont.Hand Drawn Calculator Drawn

The calculator doesn’t draw the asymptotes!!!!

Point (Removable) Discontinuities – Holes (3rd Column)

• Gaps in the graph at a single point

• Occur when • Always in the form x =• Represented by an open circle (or hole) in

the graph

0

0y

Holes, cont.Hand Drawn

2

42

x

x)x(f

Graphing Calculators and Holes• Graphing calculators have difficulty showing

removable discontinuities****Check the table for errors!

Example #1

• x-int:

• y-int:

• VA:

• HA:

• Holes:

Example #2

• x-int:

• y-int:

• VA:

• HA:

• Holes:

Your Turn:

• Complete problems 1 – 6 on the Identifying Features of Rational Equations Practice handout.

• Don’t answer the domain and range questions!

Discontinuities and Domain and Range

• Discontinuities affect the domain and range of a rational equation

• Vertical Asymptotes → Domain• Horizontal Asymptotes → Range• Holes → Domain and Range

Example 1:

•

• Domain:

• Range:

3

2

xx

y

Example 2:

•

• Domain:

• Range:

2

42

x

xy

Your Turn:

• Answer the domain and range questions for problems 1 – 6 on the Identifying Features of Rational Equations Practice handout.

Homework

• Complete problems 1 – 8 on the Identifying the Features of Rational Equations handout.