RATIONAL EXPRESSIONS AND FUNCTIONS CHAPTER 12

INTRODUCTION  In this chapter we will examine the various aspects of working with rational expressions.  We start by looking at the inverse variation.  We then examine the graphing, simplifying, and combining of rational expressions and apply this to problem solving.  We will then learn to divide polynomials and solve rational equations.

INVERSE VARIATION (12.1)  With the inverse variation, the variables will change in different directions.  The graphs of the inverse variations have the same general shape and shows the relationship between the variables.

INVERSE VARIATION (12.1)  The constant of variation will affect the shape of the inverse variation graphs.  If we know the values of x and y for one point on the graph of an inverse variation, we can find the constant of variation k and the equation of the inverse variation.

INVERSE VARIATION (12.1) Sample Problem Suppose y varies inversely with x and y = 7 when x = 5. Write an equation for the inverse variation.

INVERSE VARIATION (12.1)  Suppose (x 1, y 1 ) and (x 2,y 2 ) are two ordered pairs of an inverse variation.  Each ordered pair will have the same product constant k.  The following is also true: (x 1 )(y 1 ) = k and (x 2 )(y 2 ) = k  Therefore: (x 1 )(y 1 ) = (x 2 )(y 2 )  The last of the relationship will allow us to find missing coordinates of an inverse variation. Sample Problem The points (3,8) and (2,y) are two points on the graph of an inverse variation. Find the missing value.

INVERSE VARIATION (12.1) Sample Problem (Applying inverse variations.) The weight needed to balance a lever varies inversely with the distance from the fulcrum to the weight. Where should a student sit, who weighs 150 lbs., to balance out another student who weighs 120 lbs. and is sitting 6 ft. from he fulcrum?

INVERSE VARIATION (12.1)  When placed side-by-side we can see the difference between the inverse variation and the direct variation we learned about earlier. y varies inversely with x. y is inversely proportional to x. The product xy is constant.

INVERSE VARIATION (12.1) Sample Problem Do the data in each table represent a direct variation or an inverse variation? For each table, write an equation to model the data. a) b) XY XY

GRAPHING RATIONAL FUNCTIONS (12.2)

 The graph approaches both axes but does not cross either axes.  A line is an asymptote of a graph if the graph of the function gets closer to the line as x or y gets larger in absolute value.  In this graph the x-axis and the y-axis themselves are the asymptotes.

GRAPHING RATIONAL FUNCTIONS (12.2)  The graphs of many rational functions are related to each other.

GRAPHING RATIONAL FUNCTIONS (12.2)  Finding the vertical asymptote:  Make sure the rational function is in its simplest form, that is the denominator and numerator have no other factors in common other than 1.  The vertical asymptote will be located at the position that will make the expression in the denominator equal to zero.  That is a rational function will have a vertical asymptote whenever the denominator is zero.  The domain of a function does not include this particular x- value because that would mean division by zero.  The domain function will not include the x-value where there is a vertical asymptote because the graph of the function will approach but never touch this line.

GRAPHING RATIONAL FUNCTIONS (12.2)

 The graphs of many rational functions are related to each other.

GRAPHING RATIONAL FUNCTIONS (12.2)

The next three slide will offer a summary of the graphs of all the functions we have examined in this course.  It will be useful to keep these in mind for later courses (i.e. AP Calculus)

GRAPHING RATIONAL FUNCTIONS (12.2) Linear Function y = mx + b

GRAPHING RATIONAL FUNCTIONS (12.2) Quadratic Function y = ax 2 + bx + c

GRAPHING RATIONAL FUNCTIONS (12.2)

SIMPLIFYING RATIONAL EXPRESSIONS (12.3)

MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS (12.4)

DIVIDING POLYNOMIALS (12.5)  To divide a polynomial by a monomial, divide each term of the polynomial by the monomial divisor. Sample Problem Divide 8x 3 + 4x 2 – 12x by 2x 2.

DIVIDING POLYNOMIALS (12.5)

 When the dividend is in standard and a power is missing, add a term of that power with zero as its coefficient.  This zero coefficient serves as a “place-holder” for the long division. Sample Problem Divide (4b 3 + 5b – 3) by (2b – 1)

DIVIDING POLYNOMIALS (12.5)  In order to use the long-division method for dividing polynomials, write the divisor and/or dividend in standard form before dividing. Sample Problem Divide –3x x 2 by 1 + 3x

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS (12.6)

 To add or subtract rational expressions with different denominators, we can write the expressions with the least common denominator (LCD), which is the least common multiple (LCM) of the denominators.  We can also find the LCD of rational expressions that have polynomials with two or more terms in the denominator. LCM of Whole Numbers 4: 4, 8, 12, 16, 20 6: 6, 12, 18, 24 LCM = 12 LCM of Variable Expressions 4x: 4, 8, 12, 16, 20 6x 2 : 6, 12, 18, 24 LCM = 12x 2

ADDING AND SUBTRACTING RATIONAL EXPRESSIONS (12.6)

SOLVING RATIONAL EQUATIONS (12.7)

RATIONAL EXPRESSIONS AND FUNCTIONS CHAPTER 12 THE END