Factoring and Completing the Square: CCSS.Math.Content.HSF-IF.C.8a - Common Core: High School - Functions

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, factoring by way of completing the square falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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.

When it comes to finding equivalent forms of quadratics, there are two main approaches.

I. Factoring

II. Completing the square

This particular question wants the question to be solved using method II. completing the square. It is important to recall that the zeros of a function are areas where the graph crosses the -axis. In other words, finding the roots of a function is to find which values result in equalling zero.

For this particular problem the steps are as follows.

Step 1: Identify mathematically how completing the square works.

Given a function,

Divide the term by two, then square it and add it to both sides of the equation.

Assuming ,

Then the factored form becomes,

Recall that are constants.

Step 2: Solve for .

Apply the above steps to this particular problem to solve.

Step 1: Identify mathematically how completing the square works.

Simplifying results in,

Then the factored form becomes,

Step 2: Solve for .

Step 3: Verify results and check for extraneous solutions.

Use opposite operations to move the constants from one side to the other.

Recall the opposite operation of a squared sign is the square root sign.

Taking the square root of a number results in two values, one positive and one negative.

Step 3: Verify results.

To verify that these two values are the roots of the function, substitute them in for in the original function. If they result in zero as the output value then they are in fact a zero (root). When both values are substituted into the function and solved using a calculator it is seen that both values result in a root.

This question is testing one's ability to analyze a function algebraically and graphically and identify the vertex of a function. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, using factoring falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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: Recognize the form that the function is given in.

Recall the standard form of a quadratic is,

.

For this particular function

It is important to recall that the vertex of a parabola is either the valley or peak of a function depending on the sign in front of the squared term. If the squared term is positive then the parabola opens up and thus the vertex is a valley and if the squared term is negative then the parabola opens down and thus the vertex is a peak of the function.

Step 2: Recognize the formula for calculating the vertex of a parabola.

The coordinate of the vertex can be found using the following formula.

Substituting in the values for this particular function results in the following.

Step 3: Once the coordinate is calculated, substitute it into the original equation to solve for the coordinate of the vertex.

Therefore the vertex occurs at the point .

Step 4: Verify the vertex by graphing the function.

The above graph verifies the solution.

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, using factoring falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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: Recognize the form that the function is given in.

Recall the standard form of a quadratic is,

.

Step 2: Recognize the factored form of the original function.

Recall the factored form of a quadratic is,

where and are factors of and and are factors of for which,

Step 3: Very result.

Using the above steps for this particular problem looks as follows.

Step 1: Recognize the form that the function is given in.

therefore,

Step 2: Recognize the factored form of the original function.

First, identify the factors of and .

From here find the factors of that when added together result in .

Therefore the factored form of the function would be as follows.

Step 3: Verify result.

To verify that the factored form is equivalent to the original function, multiply the binomials together using the distributive property of each term to each other. To accomplish this multiply the first terms together, then multiply the outer terms together, then the inner terms, and finally the last terms. Once the multiplication has occurred, combine like terms to simplify.

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, factoring by way of completing the square falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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.

When it comes to finding equivalent forms of quadratics, there are two main approaches.

I. Factoring

II. Completing the square

This particular question wants the question to be solved using method II. completing the square. It is important to recall that the zeros of a function are areas where the graph crosses the -axis. In other words, finding the roots of a function is to find which values result in equalling zero.

For this particular problem the steps are as follows.

Step 1: Identify mathematically how completing the square works.

Given a function,

Divide the term by two, then square it and add it to both sides of the equation.

Assuming ,

Then the factored form becomes,

Recall that are constants.

Step 2: Solve for .

Apply the above steps to this particular problem to solve.

Step 1: Identify mathematically how completing the square works.

Simplifying results in,

Then the factored form becomes,

Step 2: Solve for .

Step 3: Verify results and check for extraneous solutions.

Use opposite operations to move the constants from one side to the other.

Recall the opposite operation of a squared sign is the square root sign.

Taking the square root of a number results in two values, one positive and one negative.

Step 3: Verify results.

To verify that these two values are the roots of the function, substitute them in for in the original function. If they result in zero as the output value then they are in fact a zero (root). When both values are substituted into the function and solved using a calculator it is seen that both values result in a root.

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, using factoring falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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: Recognize the form that the function is given in.

Recall the standard form of a quadratic is,

.

Step 2: Recognize the factored form of the original function.

Recall the factored form of a quadratic is,

where and are factors of and and are factors of for which,

Step 3: Verify result.

Using the above steps for this particular problem looks as follows.

Step 1: Recognize the form that the function is given in.

therefore,

Step 2: Recognize the factored form of the original function.

First, identify the factors of and .

From here find the factors of that when added together result in .

Therefore the factored form of the function would be as follows.

Step 3: Verify result.

To verify that the factored form is equivalent to the original function, multiply the binomials together using the distributive property of each term to each other. To accomplish this multiply the first terms together, then multiply the outer terms together, then the inner terms, and finally the last terms. Once the multiplication has occurred, combine like terms to simplify.

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, using factoring falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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: Recognize the general form of the function.

This is known as the difference of squares.

Step 2: Identify what is known.

Step 3: Substitute the known values into the difference of perfect squares found in step 1.

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, using factoring to locate the zeros of a function falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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: Recognize the general form of the function.

This is known as the difference of squares.

Step 2: Identify what is known.

Step 3: Substitute the known values into the difference of perfect squares found in step 1.

Step 4: Find the zeros of the function by setting each binomial equal to zero and solving for .

This question is testing one's ability to analyze a function algebraically and graphically and identify the vertex of a function. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, finding the line of symmetry falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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 technology to graph the function.

Step 2: Recognize the formula for calculating the vertex of a parabola.

The coordinate of the vertex can be found using the following formula.

Substituting in the values for this particular function results in the following.

Step 3: Verify by graphing that the value in the function's vertex represents the line of symmetry.

Therefore, the line of symmetry occurs at .

This question is testing one's ability to analyze a function algebraically and graphically and identify the vertex of a function. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, finding the line of symmetry falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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 technology to graph the function.

Step 2: Recognize the formula for calculating the vertex of a parabola.

The coordinate of the vertex can be found using the following formula.

Substituting in the values for this particular function results in the following.

Step 3: Verify by graphing that the value in the function's vertex represents the line of symmetry.

Therefore, the line of symmetry occurs at .

This question is testing one's ability to analyze a function algebraically and graphically and identify the vertex of a function. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, finding the line of symmetry falls within the Cluster C of analyze functions using different representations concept (CCSS.MATH.CONTENT.HSF-IF.C.8).

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 technology to graph the function.

Step 2: Recognize the formula for calculating the vertex of a parabola.

The coordinate of the vertex can be found using the following formula.

Substituting in the values for this particular function results in the following.

Step 3: Verify by graphing that the value in the function's vertex represents the line of symmetry.

Therefore, the line of symmetry occurs at .

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

For the purpose of Common Core Standards, factoring by way of completing the square falls within the Cluster C of analyze functions using different representations concept (CCSS.Math.content.HSF-IF.C.8).

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.

When it comes to finding equivalent forms of quadratics, there are two main approaches.

I. Factoring

II. Completing the square

This particular question wants the question to be solved using method II. completing the square. It is important to recall that the zeros of a function are areas where the graph crosses the -axis. In other words, finding the roots of a function is to find which values result in equalling zero.

For this particular problem the steps are as follows.

Step 1: Identify mathematically how completing the square works.

Given a function,

Divide the term by two, then square it and add it to both sides of the equation.

Assuming ,

Then the factored form becomes,

Recall that are constants.

Step 2: Solve for .

Apply the above steps to this particular problem to solve.

Step 1: Identify mathematically how completing the square works.

Simplifying results in,

Then the factored form becomes,

Step 2: Solve for .

Step 3: Verify results and check for extraneous solutions.

Use opposite operations to move the constants from one side to the other.

Step 3: Verify results.

To verify that these two values are the roots of the function, substitute them in for in the original function. If they result in zero as the output value then they are in fact a zero (root). When both values are substituted into the function and solved using a calculator it is seen that both values result in a root.