IM Alg2.2.19 Lesson: End Behavior of Rational Functions

Complete all three representations of the polynomial division following the forms of the integer division.
[size=150]In 2000, the Environmental Protection Agency (EPA) reported a combined fuel efficiency for cars that assumes 55% city driving and 45% highway driving. The expression for the combined fuel efficiency of a car that gets [math]x[/math] mpg in the city and [math]h[/math] mpg on the highway can be written as [math]\frac{100xh}{55x+45h}[/math].[/size][br][br]Several conventional cars have a fuel economy for highway driving is that is about 10 mpg higher than for city driving. That is, [math]h=x+10[/math]. Write a function [math]f[/math] that represents the combined fuel efficiency for cars like these in terms of [math]x[/math].[br]
Rewrite [math]f[/math] in the form [math]q(x)+\frac{r(x)}{b(x)}[/math] where [math]q(x),r(x),[/math] and [math]b(x)[/math] are polynomials.[br]
Complete the table to explore the end behavior for rational functions.
What do you notice about the end behavior of different types of rational functions?[br]
Graph y=j(x) and the line it approaches.
Under what conditions would the end behavior of the graph of a rational function approach a line that is not horizontal?[br]
Create a rational function that approaches the line [math]y=2x-3[/math] as [math]x[/math] gets larger and larger in either the positive or negative direction.

IM Alg2.2.19 Practice: End Behavior of Rational Functions

The function [math]f(x)=\frac{5x+2}{x-3}[/math] can be rewritten in the form [math]f(x)=5+\frac{17}{x-3}[/math]. What is the end behavior of [math]y=f(x)[/math]?
Rewrite the rational function [math]g(x)=\frac{x^2+7x-12}{x+2}[/math] in the form [math]g(x)=p(x)+\frac{r}{x+2}[/math], where [math]p(x)[/math] is a polynomial and [math]r[/math] is an integer.
Match each polynomial with its end behavior as x gets larger and larger in the positive and negative directions. (Note: Some of the answer choices are not used and some answer choices are used more than once.)
[size=150]Let the function [math]P[/math] be defined by [math]P(x)=x^3+2x^2-13x+10[/math]. Mai divides [math]P(x)[/math] by [math]x+5[/math] and gets:[/size][br][img]data:image/png;base64,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[/img][br]How could we tell by looking at the remainder that [math](x+5)[/math] is a factor?
For the polynomial function [math]f(x)=x^4+3x^3-x^2-3x[/math] we have  [math]f(\text{-}3)=0,f(\text{-}2)=\text{-}6,f(\text{-}1)=0[/math], [math]f(0)=0,f(1)=0,f(2)=30,f(3)=144[/math]. Rewrite [math]f(x)[/math] as a product of linear factors.
[size=150]There are many cones with a volume of [math]60\pi[/math] cubic inches. The height [math]h(r)[/math] in inches of one of these cones is a function of its radius [math]r[/math] in inches where [math]h(r)=\frac{180}{r^2}[/math].[/size][br]What is the height of one of these cones if its radius is 2 inches?[br]
What is the height of one of these cones if its radius is 3 inches?[br]
What is the height of one of these cones if its radius is 6 inches?[br]
[size=150]A cylindrical can needs to have a volume of 10 cubic inches. There needs to be a label around the side of the can. The function [math]S(r)=\frac{20}{r}[/math] gives the area of the label in square inches where [math]r[/math] is the radius of the can in inches.[/size][br][br]As [math]r[/math] gets closer and closer to 0, what does the behavior of the function tell you about the situation?[br]
As [math]r[/math] gets larger and larger, what does the end behavior of the function tell you about the situation?[br]
Match each rational function with a description of its end behavior as x gets larger and larger.

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