Unit 3, Lesson 4
End Behavior of Rational Functions
Integrated Math 3
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The function \(f(x)=\frac{5x+2}{x-3}\) can be rewritten in the form \(f(x)=5+\frac{17}{x-3}\). What is the end behavior of \(y=f(x)\)?
Rewrite the rational function \(g(x) = \frac{x^2+7x-12}{x+2}\) in the form \(g(x) = p(x)+ \frac{r}{x+2}\), where \(p(x)\) is a polynomial and \(r\) is an integer.
Match each rational function 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.)
\(p(x)=\dfrac{3}{x-1}\)
\(q(x)=\dfrac{2x}{x-1}\)
\(r(x)=\dfrac{2x+3}{x-1}\)
\(s(x)=\dfrac{2x^2+x+3}{x-1}\)
\(t(x)=\dfrac{x^3}{x-1}\)
The graph approaches \(y=2\).
The graph approaches \(y=3\).
The graph approaches \(y=2x+3\).
The graph approaches \(y=x^2+x+1\).
The graph approaches \(y=0\).
Let the function \(P\) be defined by \(P(x) = x^3 + 2x^2 - 13x + 10\). Mai divides \(P(x)\) by \(x+5\) and gets:
\(\displaystyle \require{enclose} \begin{array}{r} x^2-3x+2 \phantom{00}\\ x+5 \enclose{longdiv}{x^3+2x^2-13x+10} \\ \underline{-x^3-5x^2} \phantom{-13x+100} \\ -3x^2-13x \phantom{+100}\\ \underline{3x^2+15x} \phantom{+100} \\ 2x+10 \\ \underline{-2x-10} \\ 0 \end{array}\)
How could we tell by looking at the remainder that \((x+5)\) is a factor?
For the polynomial function \(f(x)=x^4+3x^3-x^2-3x\), we have \(f(\text-3)= 0, f(\text-2)=\text-6, f(\text-1)=0\), \(f(0)=0, f(1)=0,f(2)=30, f(3)=144\). Rewrite \(f(x)\) as a product of linear factors.
There are many cones with a volume of \(60\pi\) cubic inches. The height \(h(r)\), in inches, of one of these cones is a function of its radius \(r\), in inches, where \(h(r)=\frac{180}{r^2}\).
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 \(S(r)=\frac{20}{r}\) gives the area of the label in square inches, where \(r\) is the radius of the can in inches.
As \(r\) gets closer and closer to 0, what does the behavior of the function tell you about the situation?
As \(r\) gets larger and larger, what does the end behavior of the function tell you about the situation?
Match each rational function with a description of its end behavior as \(x\) gets larger and larger.
\(9x\)
\(\frac{9}{x}\)
\(\frac{99x}{x}\)
\(\frac{99+x}{x}\)
\(\frac{99x+9}{x}\)
\(\frac{99+9x}{x}\)
The value of the expression gets closer and closer to 0.
The value of the expression gets closer and closer to 1.
The value of the expression gets closer and closer to 9.
The value of the expression is 99.
The value of the expression gets larger and larger in the positive direction.
The value of the expression gets larger and larger in the negative direction.