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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

Study concepts, example questions & explanations for Common Core: High School - Functions

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All Common Core: High School - Functions Resources

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Example Questions

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Example Question #1 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the following function and its asymptotes.

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

Possible Answers:

Screen shot 2016 01 13 at 1.00.17 pm

$\textup{Asymptote: } y=0$

Screen shot 2016 01 13 at 12.43.56 pm

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

Screen shot 2016 01 13 at 12.58.06 pm

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

Screen shot 2016 01 13 at 12.43.56 pm

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

Screen shot 2016 01 13 at 12.58.06 pm

$\textup{Asymptote: }x=3$

Correct answer:

Screen shot 2016 01 13 at 12.43.56 pm

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

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:

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

Given the function

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

$\\x+3\neq0 \\x\neq-3$

thus a vertical asymptote is at x=-3 .

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 y=0 .

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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Example Question #2 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Screen shot 2016 01 13 at 12.43.56 pm

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

Question2

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

Screen shot 2016 01 13 at 12.43.56 pm

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

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=0$

Screen shot 2016 01 13 at 12.58.06 pm

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

Correct answer:

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.

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

Given the function

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

$x\neq0$

thus a vertical asymptote is at x=0 .

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 y=0 .

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

Question2

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Example Question #3 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

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

$\text{Asymptotes: }y=0$

Question3

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

Correct answer:

Question3

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

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.

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

Given the function

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

$\\x+2\neq0 \\x\neq-2$

thus a vertical asymptote is at x=-2 .

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 y=0 .

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

Question3

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Example Question #4 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Question4

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

Screen shot 2016 01 13 at 1.00.17 pm

$\text{Asymptotes: }x=4$

Screen shot 2016 01 13 at 12.58.06 pm

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

Question2

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

Question3

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

Correct answer:

Question4

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

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.

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

Given the function

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

$\\4-x\neq0 \\x\neq4$

thus a vertical asymptote is at x=4 .

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 y=0 .

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

Question4

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Example Question #5 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Question5

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

Question2

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

Question3

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

Question4

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

Screen shot 2016 01 13 at 12.58.06 pm

$\text{Asymptotes: }x=3$

Correct answer:

Question5

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

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.

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

Given the function

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

$\\x^2\neq0 \\x\neq0$

thus a vertical asymptote is at x=0 .

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 y=0 .

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

Question5

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Example Question #6 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Question3

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

Question2

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

Question5

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

Question4

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

Question6

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

Correct answer:

Question6

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

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.

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

Given the function

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

$\\2-x\neq0 \\x\neq2$

thus a vertical asymptote is at x=2 .

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 y=0 .

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

Question6

Report an Error

Example Question #7 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Question6

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

Question4

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

Question7

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

Question3

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

Question5

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

Correct answer:

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.

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

Given the function

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

$\\x-4\neq0 \\x\neq4$

thus a vertical asymptote is at x=4 .

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 y=0 .

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

Question7

Report an Error

Example Question #8 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Question4

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

Question6

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

Question5

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

Question7

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

Question8

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

Correct answer:

Question8

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

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.

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

Given the function

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

$\\x+6\neq0 \\x\neq-6$

thus a vertical asymptote is at x=-6 .

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 y=0 .

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

Question8

Report an Error

Example Question #9 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

$f(x)=\frac{1}{x-7}$

Possible Answers:

Question9

$\text{Asymptotes: }x=7,y=0$

Question8

$\text{Asymptotes: }x=-7,y=0$

Question6

$\text{Asymptotes: }x=7,y=0$

Question2

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

Question5

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

Correct answer:

Question9

$\text{Asymptotes: }x=7,y=0$

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.

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

Given the function

$f(x)=\frac{1}{x-7}$

$\\x-7\neq0 \\x\neq7$

thus a vertical asymptote is at x=7 .

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

Given the function

$f(x)=\frac{1}{x-7}$

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 y=0 .

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

Question9

Report an Error

Example Question #10 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Graph the function and its asymptotes.

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

Possible Answers:

Question10

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

Question8

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

Question9

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

Question6

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

Question5

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

Correct answer:

Question10

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

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.

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

Given the function

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

$\\x^2-1\neq0 \\x\neq\pm1$

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 f(0)=0 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

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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

Study concepts, example questions & explanations for Common Core: High School - Functions

All Common Core: High School - Functions Resources

Example Questions

Example Question #1 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #2 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #3 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #4 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #5 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #6 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #7 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #8 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #9 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

Example Question #10 : Graph Rational Functions, Identify Zeros, Asymptotes, And End Behavior: Ccss.Math.Content.Hsf If.C.7d

All Common Core: High School - Functions Resources

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