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

## 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

Therefore, the horizontal asymptote is at .

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

### 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.

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:

Given the function

thus a vertical asymptote is at .

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

Given the function

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.

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