Introduction to Rational Equations

2 Types of Functions Continuous Discontinuous

Continuous

Keeps going No breaks in graph Smooth

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 Activity 1.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. Say the student’s name before you throw him/her the ball!

Polynomial Monomial Binomial Trinomial Polygamy Polytheism Polydactyl Polyglot

Polynomials, cont. A polynomial is an algebraic expression that can be written in the form a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 x 1 + a 0 An equation or an expression with a single variable raised to (usually many) powers All exponents are whole numbers a n ≠ 0 (Leading Coefficient ≠ 0)

Polynomial Examples Generally a long list of variables f(x) = x 4 – 4x 3 + 2x 2 – 3x + 11 f(x) = x x 5 – 4x 3 + x – 12 But we can also have a short list of variables f(x) = x 5 + x f(x) = x 2 – 1 Or even no variables at all! f(x) = 10 f(x) = ½

Rational Equation

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

Polynomials vs. Rational Equations 7. f(x) = x 8 – 7x f(x) =

Your Turn: Be Prepared to Share!!! Complete problems 12 – 17 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 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 (1 st Column) Occur when the numerator is a non-zero # and the denominator equals zero Can never be crossed Always in the form x = Abbreviated VA

Vertical Asymptotes, cont. Hand DrawnCalculator Drawn The calculator doesn’t draw the asymptotes!!!!

Experiment Graph in your graphing calculator

Calculators and Vertical Asymptotes

Horizontal Asymptotes (2 nd Column) Occur when the degree of the denominator is ≥ the degree of the numerator Ex. Can be crossed when |x| is very small Describes the end behavior of a rational equation Always in the form y = Abbreviated HA

Horizontal Asymptotes, cont. Hand DrawnCalculator Drawn The calculator doesn’t draw the asymptotes!!!!

Point (Removable) Discontinuities – Holes (3 rd Column) Gaps in the graph at a single point – Occurs when Always in the form x = Represented by an open circle (or hole) in the graph

Holes, cont. Hand Drawn

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!

5.6.

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:

Example 2: Domain: Range:

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 – 6 on the Identifying the Features of Rational Equations Homework handout.

Exit Ticket Identify the following features of the graph on the right: – x-int. = – y-int. = – VA: – HA: – Holes: – Domain: – Range: