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All Common Core: High School - Functions Resources

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Flashcards: 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

Screen shot 2016 01 13 at 12.17.10 pm

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

Screen shot 2016 01 13 at 12.16.31 pm

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

Screen shot 2016 01 13 at 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 12.16.52 pm

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

Report an Error Copyright Notice

Flashcards: 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

Your answer:

Screen shot 2016 01 13 at 12.17.10 pm

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

Your answer:

Screen shot 2016 01 13 at 12.16.31 pm

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

Your answer:

Screen shot 2016 01 13 at 9.50.10 am

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

Your answer:

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

Flashcards: 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

Screen shot 2016 01 13 at 9.50.10 am

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

Question4

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

Screen shot 2016 01 13 at 12.16.31 pm

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

Question3

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

Question2

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

Report an Error Copyright Notice

Flashcards: 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

Your answer:

Screen shot 2016 01 13 at 9.50.10 am

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

Your answer:

Question4

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

Your answer:

Screen shot 2016 01 13 at 12.16.31 pm

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

Your answer:

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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All Common Core: High School - Functions Resources

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

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

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

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

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