This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=4-3 \\d=1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=3 \\d=1$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=3+1(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\\\d=\frac{1}{4}-\frac{3}{4} \\\\d=-\frac{1}{2}$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=\frac{3}{4} \\\\d=-\frac{1}{2}$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=\frac{3}{4}-\frac{1}{2}(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=3--2 \\d=5$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=-2 \\d=5$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=-2+5(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\\\d=\frac{3}{2}-\frac{1}{2} \\\\d=1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=\frac{1}{2} \\\\d=1$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=\frac{1}{2}+1(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=-1-3 \\d=-4$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=3 \\d=-4$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=3-4(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=-5-5 \\d=-10$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=5 \\d=-10$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=5-10(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=8-7 \\d=1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=7 \\d=1$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=7+1(n-1)$

This question is testing one's ability to identify and understand an arithmetic sequence and create the recursive function. Recall that for a function to be recursive, it depends on the previous term in the sequence. It is also important to recall that the difference in an arithmetic sequence is just a constant.

For the purpose of Common Core Standards, writing arithmetic and geometric recursive and explicit sequences, falls within the Cluster A of build a function that models a relationship between two quantities concept (CCSS.Math.content.HSF.BF.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 arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=\textup{Term}_2-\textup{Term}_1$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\ A=\textup{1}^{st} \textup{ term} \\d=\textup{difference constant} \\n=\textup{desired term}$

Step 3: Substitute known values into the form from Step 2.

Following the steps from above for this particular problem is as follows.

Step 1: Identify the arithmetic difference of the sequence.

$\displaystyle \\ \textup{Term}_1+d=\textup{Term}_2 \\d=-9--6\\d=-3$

Step 2: Identify the basic form for an arithmetic recursive sequence.

$\displaystyle f(n)=A+d(n-1)$

where 

$\displaystyle \\A=-6 \\d=-3$

Step 3: Substitute known values into the form from Step 2.

$\displaystyle f(n)=-6-3(n-1)$

This question is testing one's ability to identify the graphically transformation that algebraic manipulation to the function $\displaystyle f(x)$ creates. 

For the purpose of Common Core Standards, "Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k ." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.3). 

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 $\displaystyle f(x)$.

The function $\displaystyle f(x)$ in the graph above has a $\displaystyle y$-intercept at zero.

Step 2: Use technology to graph the new function $\displaystyle f(x)-2$

The function $\displaystyle f(x)-2$ in the graph above has a $\displaystyle y$-intercept at negative two.

Step 3: Compare and interpret the two graphs to identify the graphically effect.

When the two functions are plotted on the same graph where the original function is in blue and the shifted function is in orange is below.

Given the original function $\displaystyle f(x)=x^2$, the graphically effect $\displaystyle f(x)-2$ creates is a vertical shift down of two units.

Step 4: Answer the question.

In other words, $\displaystyle f(x)-2$ moves the original function $\displaystyle f(x)=x^2$ down two units.

This question is testing one's ability to identify the graphically transformation that algebraic manipulation to the function $\displaystyle f(x)$ creates. 

For the purpose of Common Core Standards, "Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k ." falls within the Cluster B of "Build new functions from existing functions" concept (CCSS.MATH.CONTENT.HSF-BF.B.3). 

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 $\displaystyle f(x)$.

The function $\displaystyle f(x)$ in the graph above has a $\displaystyle y$-intercept at zero.

Step 2: Use technology to graph the new function $\displaystyle f(x)+3$.

The function $\displaystyle f(x)+3$ in the graph above has a $\displaystyle y$-intercept at three.

Step 3: Compare and interpret the two graphs to identify the graphically effect.

When the two functions are plotted on the same graph where the original function is in blue and the shifted function is in orange is below.

Given the original function $\displaystyle f(x)=x^2$, the graphically effect $\displaystyle f(x)+3$ creates is a vertical shift upwards of three units.

Step 4: Answer the question.

In other words, $\displaystyle f(x)+3$ moves the original function $\displaystyle f(x)=x^2$ up three units.