This question is testing ones ability to recognize sequences as functions and identify specific entry values. 

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A). 

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: Identify the pattern of the given sequence.

Since the sequence starts with the assumed input value of zero and the given sequence values of,

the following logic statement can be created.

Let  represent the sequence value for the input value . In other words,

Looking at the values, the difference between entry one and entry zero is one. The difference between entry two and entry one is two. The difference between entry three and entry two is four. The difference between entry four and entry three is eight. Finally, the difference between entry five and entry four is sixteen. This pattern signifies that the increase between each entry is double the entry of the previous term.

Step 2: Write the logic statement for the sequence in mathematical terms.

Since  is equaled to a value other than zero, it is known that  must exist as an exponent.

 where  and  are some constants.

Step 3: Apply the function to the above sequence to verify.

This shows that 

thus,

Step 4: Use the function for the sequence to calculate the specific entry value.

This question is testing ones ability to recognize sequences as functions. It is also testing the concept of what it means for a function to be recursive. Recall that a function is recursive when it requires the repeat process to find the next term in a sequence.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.A). 

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: Identify the pattern of the given sequence.

The approach in this particular case.

        I. Use the function notation for the Fibonacci sequence. Recalling that the Fibonacci sequence is defined as adding the current term with the previous term to result in the next term.

           In mathematical terms this is,

            for all 

       II. Continuing the pattern found, adding each term until the twelfth term is found.

Step 2: Continue the pattern to find the particular term.

For this particular case we want to find the first eight terms of the sequence,

Step 3: Answer the question.

The sequence for the first eight entries

This question is testing ones ability to recognize sequences as functions. 

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A). 

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: Identify the pattern of the given sequence.

For this particular problem, the common difference needs to be calculated as it is an arithmetic sequence.

To identify the pattern or in other words, calculate the common difference, subtract the first term from the second term. Then subtract the second term from the third term. For an arithmetic sequence these two differences should be equal to one another.

Given this particular sequence the common difference is calculated as follows.

Therefore, the common difference is negative three. In other words, each term decreases by three every time.

Step 2: Use the common difference to find the missing terms in the sequence.

Step 3: Answer the question.

The next term in the sequence is .

This question is testing ones ability to recognize sequences as functions. 

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A). 

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: Identify the pattern of the given sequence.

Since the sequence starts with the assumed input value of zero and the given sequence values of,

the following logic statement can be created.

Let  represent the sequence value for the input value . In other words,

Since  is equaled to a value other than zero, it is known that  must exist as an exponent.

Step 2: Write the general formula of the sequence.

 where  and  are some constants.

Step 3: Using the formula from step 2 and the known characteristics from step 1, find the function that describes the sequence.

Since any value raised to the zero power equals one, the following function can be simplified and the constant  can be solved for.

Substituting the value for  into the function, solve for .

Since any value raised to the power of one equals the number itself, the following function can be simplified to solve for .

Substitute the value for  into the function to get the final solution.

his question is testing ones ability to recognize sequences as functions. 

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.MATH.CONENT.HSF-IF.A). 

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: Identify the pattern of the given sequence.

Since the sequence starts with the assumed input value of zero and the given sequence values of,

the following logic statement can be created.

Let  represent the sequence value for the input value . In other words,

Since  is equaled to a value other than zero, it is known that  must exist as an exponent.

Step 2: Write the general formula of the sequence.

 where  and  are some constants.

Step 3: Using the formula from step 2 and the known characteristics from step 1, find the function that describes the sequence.

Since any value raised to the zero power equals one, the following function can be simplified and the constant  can be solved for.

Substituting the value for  into the function, solve for .

Since any value raised to the power of one equals the number itself, the following function can be simplified to solve for .

Substitute the value for  into the function to get the final solution.

This question is testing ones ability to recognize sequences as functions. It is also testing the concept of what it means for a function to be recursive. Recall that a function is recursive when it requires the repeat process to find the next term in a sequence.

For the purpose of Common Core Standards, sequences fall within the Cluster A of the function and use of function notation concept (CCSS.Math.content.HSF-IF.A). 

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: Identify the pattern of the given sequence.

The approach in this particular case.

        I. Use the function notation for the Fibonacci sequence. Recalling that the Fibonacci sequence is defined as adding the current term with the previous term to result in the next term.

           In mathematical terms this is,

            for all 

       II. Continuing the pattern found, adding each term until the twelfth term is found.

Step 2: Continue the pattern to find the particular term.

For this particular case we have six terms of the sequence,

continuing the pattern results in the following.

CCSS.MATH.CONTENT.HSF.IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs. 

For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.Math.content.HSF-IF.B). 

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: Identify what the question is asking.

This particular question is asking for the  and  intercepts of the function.

Step 2: Determine the approach to solve the problem.

      I. Graphically plot the function using computer technology or graphing calculator.

         Then, find where the graph intersects the -axis and where it intersects the -axis as these correspond to the intercepts.

     II. Algebraically, find the intercepts by substituting in zero for  and solving for  (calculating the -intercept) and then substituting in zero for  and solving for  (calculating the -intercept).

Step 3: Choose an approach from Step 2 and perform the necessary actions.

For the purpose of this question let's solve using option II.

To algebraically solve for the -intercept substitute zero in for  and solve for . (Recall that )

Perform algebraic operations to combine like terms.

Isolate the variable on one side of the equation by using the opposite operation to move all other constants to the other side.

To algebraically solve for the -intercept substitute zero in for  and solve for .  (Recall that )

Step 4: Answer question.

Using the algebraic approach, the  and  intercepts were found to be as follows.

This question is testing one's ability to grasp the relationship between the image a function creates graphically and the intervals where the function is increase, decreasing, positive, or negative. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs. 

For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B). 

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: Identify what the question is asking.

This particular question is asking for where the function is increasing. It is important to understand that when a function is increasing, the graph exhibits a positive slope.

Step 2: Identify the intervals where the graph is positive (increasing) and where it is negative (decreasing).

Looking at the above graph, there are two intervals where the graph is increasing and one interval where it is decreasing.

As the graph approaches  from the left, the  values increase. This means that the slope for this section of the graph is positive or increasing; therefore it is one of the intervals where the function  is increasing.

Between the  values of  the   values decrease. This means that the slope for this section of the graph is negative or decreasing. 

From the  value  to infinity, the  values increase. This means that the slope for this section of the graph is also positive or increasing; therefore it is another one of the intervals where the function  is increasing.

Step 3: Answer the question.

 is increasing on the intervals .

This question is testing one's ability to grasp the relationship between a function algebraically, and the image it creates graphically. Problems like these are considered modeling problems because of their application. For example, the intercepts, extreme values, slope, symmetry, and end behavior for these functions mark key relationships between the inputs and the resulting outputs. 

For the purpose of Common Core Standards, application of interpreting functions fall within the Cluster B of the function and use of function notation concept (CCSS.MATH.CONTENT.HSF-IF.B). 

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: Identify what the question is asking.

This particular question is asking for the axis of symmetry for this particular function.

Step 2: Determine the approach to solve the problem.

      I. Graphically plot the function using computer technology or graphing calculator.

         Then, find the vertical line that splits the graph into two mirrored images. In other words, find the vertical line that is equal to the  value of the vertex of the function.

Step 3: Graph the parabola and plot the vertical line that is equal to the  value of the vertex.

Step 4: Answer the question.

The vertex is located at the point  therefore, the axis of symmetry is at .