Precalculus : Graphs of Polynomial Functions

Study concepts, example questions & explanations for Precalculus

varsity tutors app store varsity tutors android store

Example Questions

← Previous 1

Example Question #1 : Graphs Of Polynomial Functions

For what values of  will the given polynomial pass through the x-axis if plotted in Cartesian coordinates? 

Possible Answers:

None of the other answers

Correct answer:

Explanation:

One can remember that if the multiplicity of a zero is odd then it passes through the x-axis and if it's even then it 'bounces' off the x-axis. You can think about this analytically as well. What happens when we plug a number into our function just slightly above or below a zero with an even multiplicity? You find that the sign is always positive. Whereas a zero with an odd multiplicity will yield a positive on one side and a negative on the other. For zeros with odd multiplicity this alters the sign of our output and the function passes through the x-axis. Whereas the zero with even multiplicity will output a number with the same sign just above and below its zero, thus it 'bounces' off the x-axis.

Example Question #1 : Graphs Of Polynomial Functions

For this particular question we are restricting the domain of both  to nonnegative values, or the interval .

Let  and .  

For what values of  is ?

Possible Answers:

Correct answer:

Explanation:

The cubic function will increase more quickly than the quadratic, so the quadratic function must have a head start.  At , both functions evaluate to 8.  After than point, the cubic function will increase more quickly.

The domain was restricted to nonnegative values, so this interval is our only answer.

Example Question #1 : Graph A Polynomial Function

 

Which of the following is an accurate graph of ?

Possible Answers:

Varsity1

Varsity10

Varsity2

Varsity11

Varsity12

Correct answer:

Varsity1

Explanation:

is a parabola, because of the general  structure.  The parabola opens downward because .  

Solving tells the x-value of the x-axis intercept;

The resulting x-axis intercept is: .

Setting  tells the y-value of the y-axis intercept;

    

    

    

The resulting y-axis intercept is:

Example Question #1 : How To Graph An Exponential Function

Give the -intercept of the graph of the function

Round to the nearest tenth, if applicable.

Possible Answers:

The graph has no -interceptx

Correct answer:

Explanation:

The -intercept is , where :

The -intercept is .

Example Question #91 : High School: Functions

Graph the following function and identify the zeros.

Possible Answers:

Screen shot 2016 01 13 at 12.16.31 pm

Screen shot 2016 01 13 at 9.55.24 am

 

Screen shot 2016 01 13 at 9.50.10 am

Screen shot 2016 01 13 at 12.16.52 pm

Screen shot 2016 01 13 at 12.17.10 pm

Correct answer:

Screen shot 2016 01 13 at 9.55.24 am

 

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros 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 technique to factor the function.

Separating the function into two parts...

Factoring a negative one from the second set results in...

Factoring out  from the first set results in...

The new factored form of the function is,

.

Now, recognize that the first binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Screen shot 2016 01 13 at 9.55.24 am

 

Example Question #1 : Graph A Polynomial Function

Graph the function and identify the roots.

Possible Answers:

Question6

Question12

Question3

Question2

Question5

Correct answer:

Question12

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros 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 technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

        

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Question12

Example Question #1 : Graph A Polynomial Function

Graph the function and identify its roots.

Possible Answers:

Question4

Screen shot 2016 01 13 at 12.16.31 pm

Question3

Question6

Question5

Correct answer:

Question6

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros 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 technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

        

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Question6

Example Question #92 : High School: Functions

Graph the function and identify its roots.

Possible Answers:

Question2

Screen shot 2016 01 13 at 12.16.52 pm

Screen shot 2016 01 13 at 12.16.31 pm

Question3

Question4

Correct answer:

Question4

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros 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 technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

        

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Question4

Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Which could be the equation for this graph?

Polynomial

Possible Answers:

Correct answer:

Explanation:

This graph has zeros at 3, -2, and -4.5. This means that , , and . That last root is easier to work with if we consider it as and simplify it to . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying , which results in .

Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

Varsity8

Possible Answers:

Correct answer:

Explanation:

Because there are no x-intercepts, use the form , where vertex  is , so , , which gives

          

          

          

          

      

← Previous 1
Learning Tools by Varsity Tutors