Chapter 9Rational Functions

Review: Direct Variation

9-1 Inverse Variation

warm up

Joint Variation

9-3 Rational Functions and their Graphs

9-3 Rational Functions and their Graphs

9-3 Rational Functions and their Graphs

9-3 Rational Functions and their Graphs

9-3 Rational Functions and their Graphs

9-3 Rational Functions and their Graphs

9-3 Rational Functions and their GraphsTo sketch the graph of a rational function:• Determine if the function points of discontinuity for the denominator and if they are holes or vertical asymptotes. Sketch in any vertical asymptotes.•Determine if the function has a horizontal asymptote. As x gets larger (positive or negative) the graph will approach this line.•Calculate values of y for x values that are near the asymptotes. Plot these points and sketch the graph.

9-3 Rational Functions and their Graphs

9-4 Rational Expressions

9-4 Rational Expressions

9-4 Rational Expressions

9-4 Rational Expressions

9-4 Rational Expressions

9-4 Rational Expressions

9-4 Rational Expressions

9-5 Adding and Subtracting Rational Expressions

Complex Fractions

9-6 Solving Rational Equations

When a rational equation has a sum or difference of two rational expressions, you can use the LCD to simplify.

9-6 Solving Rational Equations

9-6 Solving Rational Equations

Homework: page 532 (1-21) odd

Chapter 9 test Tuesday 4/9 or Wednesday 4/10

9-6 Solving Rational Equations

Direct and Inverse variation: If y/x is always equal to the same number,

then x and y represent a direct variation. y=kx If xy is always the same value then x and y

vary inversely y = k/x

Review:

Discontinuities In rational functions discontinuities occur

where values of the variable make the denominator equal to zero.

If this value makes the numerator zero there will be a hole in the graph.

If the value does not make the numerator zero there will be a vertical asymptote in the graph.

Review:

Horizontal Asymptotes Horizontal Asymptotes describe end behavior of

graph. Determined by the degree of the functions in the

numerator and denominator. If degree in denominator is higher, horizontal

asymptote at y=0 (X axis) If degree in numerator is higher there is no

horizontal asymptote If degree is the same, horizontal asymptote

occurs at the ratio of the leading coefficients of the numerator and denominator.

Review:

Simplify Rational Expressions Factor all parts of rational expression

completely. Cancel factors that appear in both

numerator and denominator. To multiply: factor and simplify before

multiplying. To divide: Factor, flip second function,

simplify and multiply.

Review:

Adding or Subtracting Rational Expressions Must find a common denominator before

you can add or subtract.

Complex Rationals: Multiply top and bottom of rational

expression by the Least Common Multiple of all complex denominators

Review:

Solving Rational Equations If possible cross multiply to solve equations. Determine Least Common Multiple of all

rationals and multiply all terms by the LCM. Always check all of your solutions.

Review: