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 .

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 end behavior of the function. This means we will need to examine the graph of the function as the  values get larger and larger.

Step 2: Determine the approach to solve the problem.

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

         Then, find the end behavior,  values as  the  values get larger and larger.

Step 3: Interpret the above graph for the end behavior.

Looking at the graph above there appears to be a vertical asymptote and a horizontal asymptote. The vertical asymptote effects the behavior of the graph as the  values get close to zero. The horizontal asymptote effects the behavior of the graph as the  values gets closer to positive infinity and negative infinity. In other words the horizontal asymptote effects the end behavior of the function. As the  values get larger and larger the function values or end behavior of the function approaches zero.

Step 4: Verify the solution algebraically.

Using the algebraic technique, plug in larger values for  to see the trend in the function values.

As ,  since by definition, one divided by a large number becomes an extremely small number close to zero.

Step 5: Answer question.

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 periodicity of the function

Step 2: Determine the approach to solve the problem.

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

      II. Algebraically solve using the general formula,  where 

Step 3: Using option II, the algebraic method to solve for the periodicity of the given function.

Therefore the periodicity is 

.

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 decreasing. It is important to understand that when a function is decreasing, the graph exhibits a negative slope.

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

Looking at the graph above, it is seen that  is negative or decreasing roughly between . Since the specific function is also included in the question we can use algebra along with the initial assumptions made using the graph to solve the problem.

Step 3: Use algebra to find the decreasing interval of the function.

Looking at the possible answer selections, there are only two possible selections that contain two negative  values.

Option 1: 

Option 2:

Lets convert the fractions into decimals.

Since  is closer to negative one the decreasing interval of the function is,

.

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 relative minimum 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 reaches its valley.

Step 3: For the purpose of this question let's solve using graphing calculator technology.

Graphing the function results in the following graph.

Looking at the above graph the valley occurs roughly at the point .

Looking at the possible answer choices, there are only two options that contain a positive  and  value.

Option 1:

Option 2:

Since the  values are the same in these two points lets compare the  values.

 and 

Since 

 is closer to  that makes the valley of the function occur at the point, .

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 relative maximum 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 reaches its peak.

Step 3: For the purpose of this question let's solve using graphing calculator technology.

Graphing the function results in the following graph.

Step 4: Trace the function using the graphing calculator to find the coordinates of the peak.

Step 5: Answer the question.

The relative maximum of the function occurs at .

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 function 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 .