Graph Rational Functions, Identify Zeros, Asymptotes, and End Behavior: CCSS.Math.Content.HSF-IF.C.7d

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Common Core: High School - Functions › Graph Rational Functions, Identify Zeros, Asymptotes, and End Behavior: CCSS.Math.Content.HSF-IF.C.7d

Questions 1 - 10

Graph the function and its asymptotes.

$f(x)=\frac{1}{2-x}$

Question6

$\text{Asymptotes: }x=2, y=0$

Question2

$\text{Asymptotes: }x=0, y=0$

Question3

$\text{Asymptotes: }x=-2, y=0$

Question4

$\text{Asymptotes: }x=4, y=0$

Question5

$\text{Asymptotes: }x=0, y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

For the purpose of Common Core Standards, "graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

Recall that asymptotes occur at locations where the domain and range of the function do not exist. For vertical asymptotes this occurs if there is a fraction where the denominator contains the variable:

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{2-x}$

$\2-x eq0 \x eq2$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{2-x}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question6

Graph the function and its asymptotes.

$f(x)=\frac{x}{x^2-1}$

Question10

$\text{Asymptotes: }x=\pm1$

Question6

$\text{Asymptotes: }x=\pm1$

Question5

$\text{Asymptotes: }x=\pm1$

Question8

$\text{Asymptotes: }x=\pm1$

Question9

$\text{Asymptotes: }x=\pm1$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{x}{x^2-1}$

$\x^2-1 eq0 \x eq\pm1$

thus a vertical asymptote is at $x=\pm1$ .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{x}{x^2-1}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

This particular case has and has a range of all reals therefore, there is no horizontal asymptote.

Step 3: Graph the function using technology and plot the asymptotes.

Question10

Graph the function and its asymptotes.

$f(x)=\frac{2}{x-4}$

Question7

$\text{Asymptotes: }x=4,y=0$

Question3

$\text{Asymptotes: }x=4,y=0$

Question4

$\text{Asymptotes: }x=4,y=0$

Question5

$\text{Asymptotes: }x=4,y=0$

Question6

$\text{Asymptotes: }x=4,y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{2}{x-4}$

$\x-4 eq0 \x eq4$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{2}{x-4}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question7

Graph the function and its asymptotes.

$f(x)=\frac{1}{x+2}$

Question3

$\text{Asymptotes: }x=-2,y=0$

Question2

$\text{Asymptotes: }x=0,y=0$

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=-2$

Screen shot 2016 01 13 at 12.43.56 pm

$\text{Asymptotes: }x=-2,y=0$

Screen shot 2016 01 13 at 1.00.17 pm

$\text{Asymptotes: }y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{x+2}$

$\x+2 eq0 \x eq-2$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{x+2}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question3

Graph the function and its asymptotes.

$f(x)=\frac{1}{x^2}$

Question5

$\text{Asymptotes: }x=0,y=0$

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=3$

Question4

$\text{Asymptotes: }x=4,y=0$

Question3

$\text{Asymptotes: }x=-2,y=0$

Question2

$\text{Asymptotes: }x=0,y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{x^2}$

$\x^2 eq0 \x eq0$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{x^2}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question5

Graph the function and its asymptotes.

$f(x)=\frac{1}{4-x}$

Question4

$\text{Asymptotes: }x=4, y=0$

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=4, y=0$

Screen shot 2016 01 13 at 1.00.17 pm

$\text{Asymptotes: }x=4$

Question3

$\text{Asymptotes: }x=-4, y=0$

Question2

$\text{Asymptotes: }x=0, y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{4-x}$

$\4-x eq0 \x eq4$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{4-x}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question4

Graph the function and its asymptotes.

$f(x)=\frac{1}{x+6}$

Question8

$\text{Asymptotes: }x=-6,y=0$

Question4

$\text{Asymptotes: }x=-6,y=0$

Question5

$\text{Asymptotes: }x=0,y=0$

Question6

$\text{Asymptotes: }x=-2,y=0$

Question7

$\text{Asymptotes: }x=-4,y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{x+6}$

$\x+6 eq0 \x eq-6$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{x+6}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question8

Graph the function and its asymptotes.

$f(x)=\frac{1}{x+5}$

Question11

$\text{Asymptotes: }x=-5,y=0$

Question4

$\text{Asymptotes: }x=-5,y=0$

Question2

$\text{Asymptotes: }x=0,y=0$

Question8

$\text{Asymptotes: }x=-5,y=0$

Question9

$\text{Asymptotes: }x=5,y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{x+5}$

$\x+5 eq0 \x eq-5$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{x+5}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question11

Graph the function and its asymptotes.

$f(x)=\frac{1}{x}$

Question2

$\text{Asymptotes: }x=0, y=0$

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=0$

Screen shot 2016 01 13 at 12.43.56 pm

$\text{Asymptotes: }x=0, y=0$

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=0, y=0$

Screen shot 2016 01 13 at 12.43.56 pm

$\text{Asymptotes: }x=-3$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{x}$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{x}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Question2

Graph the following function and its asymptotes.

$f(x)=\frac{1}{x+3}$

Screen shot 2016 01 13 at 12.43.56 pm

$\textup{Asymptotes: }x=-3, y=0$

Screen shot 2016 01 13 at 12.43.56 pm

$\textup{Asymptote: }x=-3$

Screen shot 2016 01 13 at 12.58.06 pm

$\textup{Asymptote: }x=3$

Screen shot 2016 01 13 at 12.58.06 pm

$\textup{Asymptotes: }x=3, y=0$

Screen shot 2016 01 13 at 1.00.17 pm

$\textup{Asymptote: } y=0$

Explanation

This question is testing one's ability to graph a rational function and identify its asymptotes.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic techniques to identify asymptotes of the function.

$y=\frac{1}{a-x} \rightarrow a-x eq 0\rightarrow a eq x$

Given the function

$f(x)=\frac{1}{x+3}$

$\x+3 eq0 \x eq-3$

thus a vertical asymptote is at .

Step 2: Identify the horizontal asymptote by examining the end behavior of the function.

Given the function

$f(x)=\frac{1}{x+3}$

to find the end behavior, substitute in large values for x. When large values of x are put into the function the denominator becomes larger.

Recall that when n is some large value, the fraction approaches zero.

$\frac{1}{n}\rightarrow 0$

Therefore, the horizontal asymptote is at .

Step 3: Graph the function using technology and plot the asymptotes.

Screen shot 2016 01 13 at 12.43.56 pm

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