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Common Core: High School - Functions : High School: Functions

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

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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

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:

Question4

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

Screen shot 2016 01 13 at 9.50.10 am

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

Question2

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

Question5

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

Question3

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

Correct answer:

Question5

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

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

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

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 #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 #3 : 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:

Question7

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

Question4

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

Question2

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

Question6

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

Question5

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

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

Report an Error

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.

x^2-1

Possible Answers:

Question5

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

Question6

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

Question7

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

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

Report an Error

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.

2x^2-4

Possible Answers:

Question4

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

Question7

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

Question3

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

Question9

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

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

Report an Error

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.

x^2-49

Possible Answers:

Question5

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

Question8

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

Question7

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

Question9

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

Question10

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

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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Example Question #101 : High School: Functions

Graph the function and identify the roots.

4x^2-25

Possible Answers:

Question2

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

Question4

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

Question11

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

Question3

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

Question5

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

Correct answer:

Question11

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

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

$\\a=4\Rightarrow \sqrt{a}=\pm2 \\b=25\Rightarrow \sqrt{b}=\pm5$

thus the simplified, factored form is,

(2x-5)(2x+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.

$\\2x-5=0 \\2x=5 \\\\x=\frac{5}{2}$ $\\2x+5=0 \\2x=-5 \\\\x=-\frac{5}{2}$

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

$\begin{tabular}{ |c|c|c|c|c|c|c| } \hline x & f(x) & y \\ \hline -3& f(-3) & 11 \\ -5/2 & f(-5/2) & 0 \\ 0&f(0)& -25\\ 5/2&f(5/2)&0\\3&f(3)&11 \\\hline \end{tabular}$

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

$\\f(-3)=4(-3)^2-25=4(9)-35=36-25=11 \\f(-5/2)=4(-5/2)^2-25=4(25/4)-25=0 \\f(0)=4(0)^2-25=0-25=-25 \\f(5/2)=4(5/2)^2-25=4(25/4)-25=0 \\f(3)=4(3)^2-25=4(9)-25=36-25=11$

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

Question11

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 #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 12.58.06 pm

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

Screen shot 2016 01 13 at 1.00.17 pm

$\textup{Asymptote: } y=0$

Screen shot 2016 01 13 at 12.58.06 pm

$\textup{Asymptote: }x=3$

Screen shot 2016 01 13 at 12.43.56 pm

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

Screen shot 2016 01 13 at 12.43.56 pm

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

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

Report an Error

Example Question #1 : 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 12.58.06 pm

$\text{Asymptotes: }x=0$

Screen shot 2016 01 13 at 12.58.06 pm

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

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

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

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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All Common Core: High School - Functions Resources

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

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Common Core: High School - Functions : High School: Functions

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

All Common Core: High School - Functions Resources

Example Questions

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

Example Question #3 : 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 #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 #101 : High School: Functions

Example Question #2 : Graph A Polynomial Function

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

Example Question #1 : 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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