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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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Example Question #11 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c

Graph the function and identify the roots.

4x^2-25

Possible Answers:

Question4

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

Question2

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

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.

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

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

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

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Example Question #4 : Graph A Polynomial Function

Graph the function and identify the roots.

100x^2-1

Possible Answers:

Question6

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

Question5

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

Question2

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

Question3

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

Question12

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

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

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

Example Question #4 : Graph A Polynomial Function

All Common Core: High School - Functions Resources

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