Types of Variation1) Direct Variation: “y varies directly with x”:

where .Plot : on the TI-nspire

2) Inverse Variation: “y varies inversely x”: where .Plot: on the TI-nspire

3) Joint variation: “ z varies jointly with x and y”: where . (Note: z varies directly with both x and y.)

How can we Identify an Inverse Variation?

• Note: implies that : • Does the following data satisfy an inverse

relationship?

• Does a plot of the points suggest an inverse relationship?

• Does the data obey the relationship:

x 2 4 10 15

y 15 7.5 3 2

ExamplesIdentify the following data sets as examples of direct or inverse variation, or neither.

3)

x 2 4 10 15

y 10 8 3 1.5

x 0.2 0.5 1.0 1.5

y 8 20 40 60

x 0.2 0.5 1.0 2.0

y 40. 16 8.0 4.0

Rational Functions• A rational function can be written as a ratio of two

polynomial functions:

• A rational function is in simplest form when P(x) and Q(x) have no common divisors.• When simplifying a rational expression, any restrictions on the domain of the original expression carry over to the simplified form. Ex: with • So note: the restriction carries over to the simplified form.

Examples• Simplify the following:1) =

2) =

3) =

4) =

Multiplying Rational Expressions

• Works in the same was fractions: Multiply the numerators to get the new numerator and multiply denominators to get the new denominator. Examples:

1) =

Dividing Rational Expressions

• Since: =

We multiply the dividend by the reciprocal of the divisor. (All denominators encountered restrict the domain) In Short:

Examples

1) =

Adding and Subtracting Rational Expressions

• First you need to get each expression to have the same denominator.

• Then add/subtract the numerators using the common denominator.

• It’s usually easiest to 1. simplify the original denominators so as to see any

common factors they share. 2. Then multiply top and bottom of each expression by

the factors not shared by their denominators until the denominators are the same.EXAMPLES FOLLOW

Examples1) =

2) =

Simplifying Complex Fractions• A complex fraction is an expression in which the

numerator and/or the denominator contains a fraction.

• We first saw it in Physics with the units of acceleration: which simplified to : .

• Multiply all fractions within the complex fraction by their lowest common denominators and simplify.

• Examples follow.

Examples

1) =

2) =

How Can we Solve Rational Equations?• A rational equation contains at least one rational

expression. To solve it:First, get rid of the denominators by multiplying through

by their prime factors. This process can introduce extraneous solutions.

Next simplify and solve for the unknown.Only solutions that obey original restrictions on the

domain are allowable.Finally, check for extraneous solutions by substituting

your candidate solutions into the original equation.OR: Graph both sides of the equation and use the

INTERSECT function. You must still check the validity of the solutions.

ExamplesSolve:

More Examples1) A plane flies a roundtrip of 3700 miles in 7 hours.

Without wind, the plane cruises at 480 . However, the above trip involves an east to west wind which speeds up eastward flights and slows westward flights. Find the wind speed.

2) Solve using a graphing calculator:

Solving Systems of Rational Equations

We can solve systems of rational equations using substitution:

Solve for in each equation, then set the expressions involving equal to each other, and solve for .

Examples

1)

Solving Rational Inequalities1) Write the inequality as an equation and solve it.

2) Find the forbidden values.

3) Use the values obtained from (1) and (2) to create intervals on a number line.

4) Then test points within each interval to see if the inequality condition is observed. Report the good intervals in the usual interval notation.

OR GRAPH each side and choose the correct region.

Examples1)

2)

Hwk 41• Pages 546-548: 19, 30, 32, 34, 37, 49, 50• Page 551: 14

Hwk 41• Pages 546-548: 19, 30, 32, 34, 37, 49, 50• Page 551: 14

Hwk 41• Pages 546-548: 19, 30, 32, 34, 37, 49, 50Page 551: 14

Hwk 41• Pages 546-548: 19, 30, 32, 34, 37, 49, 50• Page 551: 14