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Common Core: High School - Functions : Graph Polynomial Functions, Identify Zeros, Factor, and Identify End Behavior.: CSS.Math.Content.HSF-IF.C.7c

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All Common Core: High School - Functions Resources

6 Diagnostic Tests 82 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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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 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 9.50.10 am

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

Screen shot 2016 01 13 at 12.16.52 pm

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

Screen shot 2016 01 13 at 9.55.24 am

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

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

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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.

x^2-4

Possible Answers:

Screen shot 2016 01 13 at 9.50.10 am

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

Question2

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

Question4

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

Screen shot 2016 01 13 at 12.16.31 pm

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

Question3

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

Correct answer:

Question2

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

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

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

since

$b=4\Rightarrow \sqrt{b}=\pm2$

thus the simplified, factored form is,

(x-2)(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.

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

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -2& f(-2) & 0 \\ -1 & f(-1) & -3 \\ 0&f(0)& -4\\ 1&f(1)&-3\\2&f(2)&0 \\ 3&f(3)&5 \\\hline \end{tabular}$

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

$\\f(-2)=x^2-4=(-2)^2-4=4-4=0 \\f(-1)=(-1)^2-4=1-4=-3 \\f(0)=0^2-4=-4 \\f(1)=1^2-4=1-4=-3 \\f(2)=2^2-4=4-4=0 \\f(3)=3^2-4=9-4=5$

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

Question2

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Example Question #2 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify the roots.

x^2-9

Possible Answers:

Question2

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

Question3

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

Question4

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

Screen shot 2016 01 13 at 12.16.52 pm

$\text{Root: }x=-1$

Screen shot 2016 01 13 at 9.50.10 am

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

Correct answer:

Question3

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

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

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

since

$b=9\Rightarrow \sqrt{b}=\pm3$

thus the simplified, factored form is,

(x-3)(x+3) .

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.

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

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -3& f(-3) & 0 \\ -1 & f(-1) & -8 \\ 0&f(0)& -9\\ 1&f(1)&-8\\3&f(3)&0 \\\hline \end{tabular}$

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

$\\f(-3)=x^2-9=(-3)^2-9=9-9=0 \\f(-1)=(-1)^2-9=1-9=-8 \\f(0)=0^2-9=-9 \\f(1)=1^2-9=-8 \\f(3)=3^2-9=9-9=0$

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

Question3

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Example Question #6 : Graph A Polynomial Function

Graph the function and identify its roots.

4x^2-36

Possible Answers:

Question4

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

Screen shot 2016 01 13 at 12.16.52 pm

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

Screen shot 2016 01 13 at 12.16.31 pm

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

Question3

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

Question2

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

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

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Example Question #3 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify its roots.

x^2-25

Possible Answers:

Question5

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

Question2

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

Question3

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

Screen shot 2016 01 13 at 9.50.10 am

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

Question4

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

Correct answer:

Question5

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

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

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

since

$b=25\Rightarrow \sqrt{b}=\pm5$

thus the simplified, factored form is,

(x-5)(x+5) .

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.

$\\x-5=0 \\x=5$ $\\x+5=0 \\x=-5$

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -5& f(-5) & 0 \\ -2 & f(-2) & -21 \\ 0&f(0)& -25\\ 2&f(2)&-21\\5&f(5)&0 \\\hline \end{tabular}$

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

$\\f(-5)=(-5)^2-25=25-25=0 \\f(-2)=(-2)^2-25=4-25=-21 \\f(0)=0^2-25=-25 \\f(2)=2^2-25=4-25=-21 \\f(5)=5^2-25=25-25=0$

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

Question5

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Example Question #5 : Graph A Polynomial Function

Graph the function and identify its roots.

9x^2-16

Possible Answers:

Question5

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

Question6

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

Question4

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

Question3

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

Screen shot 2016 01 13 at 12.16.31 pm

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

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

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Example Question #4 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify the roots.

25x^2-1

Possible Answers:

Question2

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

Question7

$\text{Roots: }x=\pm\frac{1}{5}$

Question5

$\text{Roots: }x=\pm{5}$

Question6

$\text{Roots: }x=\pm\frac{5}{2}$

Question4

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

Correct answer:

Question7

$\text{Roots: }x=\pm\frac{1}{5}$

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

$\\a=25\Rightarrow \sqrt{a}=\pm5 \\b=1\Rightarrow \sqrt{b}=\pm1$

thus the simplified, factored form is,

(5x-1)(5x+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.

$\\5x-1=0 \\5x=1\\\\x=\frac{1}{5}$ $\\5x+1=0 \\5x=-1 \\\\x=-\frac{1}{5}$

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -1& f(-1) & 24 \\ 0 & f(0) & -1 \\ 1/5&f(1/5)& 0\\ 1&f(1)&24\\\hline \end{tabular}$

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

$\\f(-1)=25(-1)^2-1=25-1=24 \\f(0)=25(0)^2-1=0-1=-1 \\f(1/5)=25(1/5)^2-1=25(1/25)-1=1-1=0 \\f(1)=25(1)^2-1=25-1=24$

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

Question7

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Example Question #5 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify the roots.

x^2-1

Possible Answers:

Question7

$\text{Roots: }x=\pm\frac{1}{5}$

Question6

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

Question5

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

Question4

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

Question8

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

Correct answer:

Question8

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

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

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

since

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

thus the simplified, factored form is,

(x-1)(x+1) .

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.

$\\x-1=0 \\x=1$ $\\x+1=0 \\x=-1$

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -2& f(-2) & 3 \\ -1 & f(-1) & 0 \\ 0&f(0)& -1\\ 1&f(1)&0\\5&f(2)&3 \\\hline \end{tabular}$

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

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

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

Question8

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Example Question #6 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify the roots.

2x^2-4

Possible Answers:

Question7

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

Question9

$\text{Roots: }x=\pm\sqrt{2}$

Question3

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

Question4

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

Question8

$\text{Roots: }x=\pm\sqrt{2}$

Correct answer:

Question9

$\text{Roots: }x=\pm\sqrt{2}$

Explanation:

his 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 techniques to solve for the roots.

$\\2x^2-4=0 \\2x^2=4 \\x^2=2 \\x=\pm\sqrt{2}$

Step 2: Create a table of (x,y) pairs.

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

Step 3: The values in the table are found by substituting in the x values into the function as follows.

$\\f(-2)=2(-2)^2-4=2(4)-4=8-4=4 \\f(-1)=2(-1)^2-4=2(1)-4=2-4=-2 \\f(0)=2(0)^2-4=0-4=-4$

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

Question9

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

Graph the function and identify the roots.

x^2-49

Possible Answers:

Question10

$\text{Roots: }x=\pm7$

Question7

$\text{Roots: }x=\pm\frac{1}{7}$

Question9

$\text{Roots: }x=\pm7$

Question8

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

Question5

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

Correct answer:

Question10

$\text{Roots: }x=\pm7$

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

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

since

$b=49\Rightarrow \sqrt{b}=\pm7$

thus the simplified, factored form is,

(x-7)(x+7) .

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.

$\\x-7=0 \\x=7$ $\\x+7=0 \\x=-7$

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -8& f(-8) & 15 \\ -7 & f(-7) & 0 \\ 0&f(0)& -49\\ 7&f(7)&-0\\8&f(8)&15 \\\hline \end{tabular}$

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

$\\f(-8)=(-8)^2-49=64-49=15 \\f(-7)=(-7)^2-49=49-49=0 \\f(0)=0^2-49=-49 \\f(7)=(7)^2-49=49-49=0 \\f(8)=(8)^2-49=64-49=15$

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

Question10

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All Common Core: High School - Functions Resources

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Common Core: High School - Functions : Graph Polynomial Functions, Identify Zeros, Factor, and Identify End Behavior.: CSS.Math.Content.HSF-IF.C.7c

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

All Common Core: High School - Functions Resources

Example Questions

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

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

Example Question #2 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Example Question #6 : Graph A Polynomial Function

Example Question #3 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Example Question #5 : Graph A Polynomial Function

Example Question #4 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Example Question #5 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Example Question #6 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

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

All Common Core: High School - Functions Resources

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