Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Graphs of Polynomial Functions

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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?

f(x)=(x+2)^2(x-1)(x+3)(x-5)^3

Possible Answers:

x= -3,1

x= -3,-2,1,5

x=-3,1,5

None of the other answers

x= -2,5

Correct answer:

x=-3,1,5

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.

Report an Error

Example Question #2 : Graphs Of Polynomial Functions

For this particular question we are restricting the domain of both $f \, \textup{and}\,g$ to nonnegative values, or the interval $\left [0, \right \infty )$ .

Let f(x)=x^2+4 and g(x) = x^3 .

For what values of is f(x) > g(x) ?

Possible Answers:

$[4,\infty)$

$[2,\infty]$

[0,2)

[0,8)

Correct answer:

[0,2)

Explanation:

The cubic function will increase more quickly than the quadratic, so the quadratic function must have a head start. At x=2 , 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.

Report an Error

Example Question #1 : Graph A Polynomial Function

Which of the following is an accurate graph of?

Possible Answers:

Varsity10

Varsity12

Varsity2

Varsity11

Varsity1

Correct answer:

Varsity1

Explanation:

f(x) is a parabola, because of the general $f(x) = x^{2}$ structure. The parabola opens downward because f(x) < 0 .

Solving $-(x + 4)^{2} = 0$ tells the x-value of the x-axis intercept;

$0 = (x + 4)^{2}$

$\sqrt{0} = \sqrt{(x + 4)^{2}}$

0 = x + 4

-4 = x

The resulting x-axis intercept is: (-4, 0) .

Setting x = 0 tells the y-value of the y-axis intercept;

= -1(16)

= -16

The resulting y-axis intercept is: (0, -16)

Report an Error

Example Question #1 : How To Graph An Exponential Function

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest tenth, if applicable.

Possible Answers:

The graph has no -interceptx

(3.65,0)

( -2.92, 0)

(2.63,0)

( -3.08, 0)

Correct answer:

(2.63,0)

Explanation:

The -intercept is (a,0) , where f(a)= 0 :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$5 \cdot 4^{a- 3}- 3 = 0$

$5 \cdot 4^{a- 3}- 3 + 3= 0 + 3$

$5 \cdot 4^{a- 3}= 3$

$5 \cdot 4^{a- 3} \div 5= 3 \div 5$

$4^{a- 3} = 0.6$

$\ln \left (4^{a- 3} \right )=\ln 0.6$

$(a- 3)\ln 4 =\ln 0.6$

$a- 3=\frac{\ln 0.6}{\ln 4 } \approx \frac{-0.5108}{1.3863}\approx -0.3685$

$a- 3+3 \approx -0.3685 + 3$

$a \approx 2.63$

The -intercept is (2.63,0) .

Report an Error

Example Question #1 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the following function and identify the zeros.

x^3-2x^2-x+2

Possible Answers:

Screen shot 2016 01 13 at 9.55.24 am

$\textup{Roots: }x=-1, x=1, x=2$

Screen shot 2016 01 13 at 9.50.10 am

$\textup{Roots: } x=1, x=2$

Screen shot 2016 01 13 at 12.16.31 pm

$\textup{Roots: }x=-1, x=0, x=2$

Screen shot 2016 01 13 at 12.17.10 pm

$\textup{Roots: }x=-1, x=2$

Screen shot 2016 01 13 at 12.16.52 pm

$\textup{Roots: }x=-1, x=1, x=-2$

Correct answer:

Screen shot 2016 01 13 at 9.55.24 am

$\textup{Roots: }x=-1, x=1, x=2$

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.

x^3-2x^2-x+2

Separating the function into two parts...

(x^3-2x^2)+(-x+2)

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

(x^3-2x^2)-(x-2)

Factoring out x^2 from the first set results in...

x^2(x-2)-1(x-2)

The new factored form of the function is,

(x^2-1)(x-2) .

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

$(x^2-b^2)=(x-\sqrt{b})(x+\sqrt{b})$

since

$b=1\Rightarrow \sqrt{b}=\pm1$

thus the simplified, factored form is,

(x-1)(x+1)(x-2) .

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.

$\\(x-1)(x+1)(x-2)=0 \\x-1=0\Rightarrow x=1 \\x+1=0\Rightarrow x=-1 \\x-2=0\Rightarrow x=2$

Step 3: Create a table of (x,y) pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -2& f(-2) & -12 \\ -1 & f(-1) & 0 \\ 0&f(0)& 2\\ 1.5&f(1.5)&-0.625\\2&f(2)&0 \\ 3&f(3)&8 \\\hline \end{tabular}$

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

$\\f(-2)=(-2)^3-2(-2)^2-(-2)+2=-8-8+2+2=-16+4=-12 \\f(-1)=(-1)^3-2(-1)^2-(-1)+2=-1-2+1+2=0 \\f(0)=(0)^3-2(0)^2-(0)+2=0-0-0+2=2 \\f(1.5)=(1.5)^3-2(1.5)^2-(1.5)+2=-0.625 \\f(2)=(2)^3-2(2)^2-(2)+2=8-8-2+2=0 \\f(3)=(3)^3-2(3)^2-(3)+2=27-18-3+2=8$

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

Report an Error

Example Question #2 : Graph A Polynomial Function

Graph the function and identify the roots.

100x^2-1

Possible Answers:

Question12

$\text{Roots: }x=\pm\frac{1}{10}$

Question2

$\text{Roots: }x=\pm\frac{1}{10}$

Question3

$\text{Roots: }x=\pm3$

Question5

$\text{Roots: }x=\pm5$

Question6

$\text{Roots: }x=\pm\frac{1}{10}$

Correct answer:

Question12

$\text{Roots: }x=\pm\frac{1}{10}$

Explanation:

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

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

$(a^2x^2-b^2)=(\sqrt{a}x-\sqrt{b})(\sqrt{a}x+\sqrt{b})$

since

$\\a=100\Rightarrow \sqrt{a}=\pm10 \\b=1\Rightarrow \sqrt{b}=\pm1$

thus the simplified, factored form is,

(10x-1)(10x+1) .

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.

$\\10x-1=0 \\10x=1\\ \\x=\frac{1}{10}$ $\\10x+1=0 \\10x=-1 \\\\x=-\frac{1}{10}$

Step 3: Create a table of (x,y) pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -1& f(-1) & 99 \\ 0 & f(0) & -1 \\ 1&f(1)& 99 \\\hline \end{tabular}$

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

$\\f(-1)=100(-1)^2-1=100-1=99 \\f(0)=100(0)^2-1=0-1=-1 \\f(1)=100(1)^2-1=100-1=99$

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

Question12

Report an Error

Example Question #3 : Graph A Polynomial Function

Graph the function and identify its roots.

9x^2-16

Possible Answers:

Question5

$\text{Roots: }x=\pm\frac{3}{4}$

Question3

$\text{Roots: }x=\pm\frac{3}{4}$

Screen shot 2016 01 13 at 12.16.31 pm

$\text{Roots: }x=\pm\frac{1}{4}$

Question4

$\text{Roots: }x=\pm\frac{4}{3}$

Question6

$\text{Roots: }x=\pm\frac{4}{3}$

Correct answer:

Question6

$\text{Roots: }x=\pm\frac{4}{3}$

Explanation:

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

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

$(a^2x^2-b^2)=(\sqrt{a}x-\sqrt{b})(\sqrt{a}x+\sqrt{b})$

since

$\\a=9\Rightarrow \sqrt{a}=\pm3 \\b=16\Rightarrow \sqrt{b}=\pm4$

thus the simplified, factored form is,

(3x-4)(3x+4) .

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.

$\\3x-4=0 \\3x=4\\\\x=\frac{4}{3}$ $\\3x+4=0 \\3x=-4 \\\\x=-\frac{4}{3}$

Step 3: Create a table of (x,y) pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -2& f(-2) & 20 \\ -4/3 & f(-4/3) & 0 \\ 0&f(0)& -16\\ 1&f(1)&-7\\\hline \end{tabular}$

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

$\\f(-2)=9(-2)^2-16=9(4)-16=36-16=20 \\f(-4/3)=9(-4/3)^2-16=16-16=0 \\f(0)=9(0)^2-10-16=-16 \\f(1)=9(1)^2-16=9-16=-7$

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

Question6

Report an Error

Example Question #1 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify its roots.

4x^2-36

Possible Answers:

Screen shot 2016 01 13 at 12.16.31 pm

$\text{Roots: }x=\pm3$

Question4

$\text{Roots: }x=\pm3$

Question2

$\text{Roots: }x=\pm2$

Question3

$\text{Roots: }x=\pm3$

Screen shot 2016 01 13 at 12.16.52 pm

$\text{Roots: }x=-1, x=2$

Correct answer:

Question4

$\text{Roots: }x=\pm3$

Explanation:

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

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

$(a^2x^2-b^2)=(\sqrt{a}x-\sqrt{b})(\sqrt{a}x+\sqrt{b})$

since

$\\a=4\Rightarrow \sqrt{a}=\pm2 \\b=36\Rightarrow \sqrt{b}=\pm6$

thus the simplified, factored form is,

(2x-6)(2x+6) .

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.

$\\2x-6=0 \\2x=6 \\x=3$ $\\2x+6=0 \\2x=-6 \\x=-3$

Step 3: Create a table of (x,y) pairs.

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -3& f(-3) & 0 \\ -1 & f(-1) & -32 \\ 0&f(0)& -36\\ 1&f(1)&-32\\3&f(3)&0 \\\hline \end{tabular}$

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

$\\f(-3)=4(-3)^2-36=4(9)-36=36-36=0 \\f(-1)=4(-1)^2-36=4-36=-32 \\f(0)=4(0)^2-36=0-36=-36 \\f(1)=4(1)^2-36=4-36=-32 \\f(3)=4(3)^2-36=4(9)-36=36-36=0$

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

Question4

Report an Error

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:

y=2x^3 -7x^2 -21x + 54

y=-2x^3 +7x^2 +21x -54

y=-2x^3 -7x^2 + 21x + 54

y=2x^3 +7x^2 -21x -54

y=2x^3-x^4+2x-1

Correct answer:

y=-2x^3 -7x^2 + 21x + 54

Explanation:

This graph has zeros at 3, -2, and -4.5. This means that x-3=0 , x+2 =0 , and x+4.5=0 . That last root is easier to work with if we consider it as $x+\frac{9}{2}=0$ and simplify it to 2x+9=0 . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying -(x-3)(x+2)(2x+9) , which results in -2x^3 -7x^2 +21x +54 .

Report an Error

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

Write the quadratic function for the graph:

Varsity8

Possible Answers:

$f(x) = -x^{2} - 6x - 12$

$f(x) = x^{2} - 6x + 12$

$f(x) = -x^{2} - 12x + 3$

$f(x) = x^{2} + 12x + 3$

$f(x) = x^{2} + 6x - 12$

Correct answer:

$f(x) = x^{2} - 6x + 12$

Explanation:

Because there are no x-intercepts, use the form $f(x) = (x - h)^{2} + k$ , where vertex (h, k) is (3, 3) , so h = 3 , k = 3 , which gives

$f(x) = (x - 3)^{2} + 3$

= (x - 3)(x - 3) + 3

$= x^{2} + 2(-3)(x) + (-3)^{2} + 3$

$= x^{2} - 6x + 9 + 3$

$= x^{2} - 6x + 12$

Report an Error

View Pre-Calculus Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Pre-Calculus Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

View Pre-Calculus Tutors

Ian
Certified Tutor

Lipscomb University, Bachelor of Science, Biology, General.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ACT Tutors in Seattle, Spanish Tutors in Chicago, Biology Tutors in Atlanta, MCAT Tutors in Miami, Statistics Tutors in Dallas Fort Worth, English Tutors in Dallas Fort Worth, ISEE Tutors in Miami, French Tutors in Houston, Computer Science Tutors in San Diego, GRE Tutors in Seattle

Popular Courses & Classes

MCAT Courses & Classes in San Diego, ISEE Courses & Classes in Washington DC, SSAT Courses & Classes in New York City, MCAT Courses & Classes in Los Angeles, LSAT Courses & Classes in New York City, GRE Courses & Classes in Miami, Spanish Courses & Classes in Philadelphia, LSAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Denver, SAT Courses & Classes in Chicago

Popular Test Prep

SSAT Test Prep in Boston, LSAT Test Prep in New York City, MCAT Test Prep in New York City, ACT Test Prep in Seattle, GRE Test Prep in San Francisco-Bay Area, LSAT Test Prep in Washington DC, MCAT Test Prep in Phoenix, MCAT Test Prep in Dallas Fort Worth, SAT Test Prep in Boston, MCAT Test Prep in Los Angeles

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Graphs of Polynomial Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Graphs Of Polynomial Functions

Example Question #2 : Graphs Of Polynomial Functions

Example Question #1 : Graph A Polynomial Function

Example Question #1 : How To Graph An Exponential Function

Example Question #1 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Example Question #2 : Graph A Polynomial Function

Example Question #3 : Graph A Polynomial Function

Example Question #1 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

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

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

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: