Interpreting Functions - Common Core: High School - Functions

This question is testing the concept and understanding of a function's domain. It is important to recall that domain can be identified graphically or algebraically. Graphically, range contains the y-values that span the image of the function where as domain, contains the x-values of the function. Algebraically, domain is known as the input values x of a function that then results in the y outputs. In other words, the input values when placed into the function results in the y values creating an (x, y) pair. It is also critical to understand that there are two important restrictions on domain. These restrictions occur when the x variable is in the denominator of a function or under a radical. This is because a fraction does not exist when there is a zero in the denominator; if under the radical is negative it results in imaginary values.

For the purpose of Common Core Standards, domain and range 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 function and what the question is asking.

Find the domain (x values) and range (y values) of the function. This includes identifying points where x values do not result in a y value. At these points the domain does not exist.

Step 2. Discuss the options to solve the problem.

I. Graphically plot the function by computer/technology resource. Then interpret the graph.

II. Create a table of (x, y) pairs and plot the points to create the graph. Then interpret the graph.

III. Algebraically find the x values that do not exist, these will be areas that are not in the domain.

For this particular function let's use the third option to find the domain.

Any time there is a fraction set the denominator equal to zero and solve for x, this will be a value that x cannot equal since it is not in the domain. Also, anytime there is a radical set the radicand equal to zero and solve for x; this will also be a value x cannot equal as it is also not in the domain.

Fractions:

Radicals

Step 3: Use algebraic technique to solve the problem.

For this particular function there exists a fraction and a radical therefore, the denominator of the fraction needs to be set to zero and solve for x. Also the radicand needs to be set to zero and solve for x.

For the fraction, using algebraic manipulations we get:

For the radicand, using algebraic manipulations we get:

Interpret the results to identify the domain. Since we found the areas where the domain does not exist we can state the domain as all real values of x except the areas for which we found to not exist. It is important to understand that "all reals" refer to all numbers negative, positive, zero, fractions, and decimal values.

Since the function is linear, the x variable is singular, we know that the range is all real y values.

Therefore, the domain and range solution in mathematical terms is as follows.

This question is testing the concept and understanding of a function's domain. It is important to recall that domain can be identified graphically or algebraically. Graphically, domain contains the x-values of the function where as range, contains the y-values that span the image of the function. Algebraically, domain is known as the input values x of a function that then results in the y outputs. In other words, the input values when placed into the function results in the y values creating an (x, y) pair. It is also critical to understand that there are two important restrictions on domain. These restrictions occur when the x variable is in the denominator of a function or under a radical. This is because a fraction does not exist when there is a zero in the denominator; if under the radical is negative it results in imaginary values.

For the purpose of Common Core Standards, domain and range 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 function and what the question is asking.

Find the domain (x values) of the function. This includes identifying points where x values do not result in a y value. At these points the domain does not exist.

Step 2. Discuss the options to solve the problem.

I. Graphically plot the function by computer/technology resource. Then interpret the graph.

II. Create a table of (x, y) pairs and plot the points to create the graph. Then interpret the graph.

III. Algebraically find the x values that do not exist, these will be areas that are not in the domain.

For this particular function let's use the third option to simplify the function and then use the first option to graph and solve for the domain.

To simplify this function factor out a common factor of in the numerator and denominator.

Any time there is a fraction set the denominator equal to zero and solve for x, this will be a value that x cannot equal since it is not in the domain.

Fractions:

Step 3: Use algebraic technique along with graphing the function to solve the problem.

Graph to verify solution.

The graph above shows the function to have a vertical asymptote at therefore it is not in the domain.

Thus the solution to this problem is,

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, 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.

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 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 particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 80 minutes. Therefore to find the units that the -axis will have, convert 80 into intervals of five minutes.

The maximum speed is 12 miles per hour. Therefore, the -axis spans from zero to 12.

At some value the function reaches a maximum value of,

.

Since the question states, "A vehicle increases its speed by miles per hour every five minutes for 80 minutes", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 90 minutes. Therefore to find the units that the -axis will have, convert 90 minutes into intervals of five minutes.

Therefore, the -axis spans from zero to 5.

At some value the function reaches a maximum value of,

.

Since the question states, "A vehicle decreases its speed by miles per hour every five minutes for 90 minutes", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 125 minutes. Therefore to find the units that the -axis will have, convert 125 minutes into intervals of five minutes.

Since the question states, "A vehicle starts to increase its speed after 25 minutes at a rate of miles per hour every five minutes for 125 minutes", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 175 minutes. Therefore to find the units that the -axis will have, convert 175 minutes into intervals of five minutes.

The maximum speed is 45 miles per hour. Therefore, the -axis spans from zero to 45.

At some value the function reaches a maximum value of,

.

Since the question states, "A vehicle starts to increase its speed after 50 minutes at a rate of miles per hour every five minutes for 175 minutes resulting in a maximum speed of 45 miles per hour", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 225 minutes. Therefore to find the units that the -axis will have, convert 225 minutes into intervals of five minutes.

Since the question states, "A vehicle decreases its speed by miles per hour every five minutes for 90 minutes", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 200 minutes. Therefore to find the units that the -axis will have, convert 200 minutes into intervals of five minutes.

Since the question states, "A vehicle starts to increase its speed after 100 minutes at a rate of miles per hour every five minutes for 200 minutes", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This particular question is testings one's ability to recognize characteristics of a function in terms of its context. Specifically, it is testing the ability to identify the domain of a function, or in other words, the possible input values that logically and mathematically work for the situation that is described by .

For the purpose of Common Core Standards, "relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes" falls within the Cluster B of "interpret functions that arise in applications in terms of the context" concept (CCSS.MATH.CONTENT.HSF-IF.B.5).

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 for.

This question is asking to find the domain, possible x values that make sense for this situation .

Step 2: Use the information that describes and create a graph that could possibly fit.

Let us identify the known information of the function.

is in intervals of "miles per every five minutes" and goes from zero to 225 minutes. Therefore to find the units that the -axis will have, convert 225 minutes into intervals of five minutes.

Since the question states, "A vehicle decrease its speed after 75 minutes at a rate of miles per hour every five minutes for 225 minutes", a linear relationship being time and speed is assumed.

Using the known information creates the graph below.

Step 3: Using the graph above, identify the domain.

Recalling that the domain of a function is the interval of values that result in a real output that lies in the range of the function. In mathematical terms,

.

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

For the purpose of Common Core Standards, "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.9).

Step 1: Identify the maximum of Table 1.

Using the table find the time value where the largest distance exists.

Now, let us plot the points from the table and connect them with a smooth curve to represent the function.

It is seen from the table and the graph that the vertex or maximum of the function exists at .

Step 2: Identify the maximum of Figure 2.

Looking at Figure 1, plotting the vertex and extending a vertical and horizontal line, we can find the coordinate pair of the vertex.

Therefore the vertex or maximum of Figure 1 is .

Step 3: Compare the maximums from step 1 and step 2.

Compare the value coordinate from both maximums.

Therefore, Figure 1 has the largest maximum.

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

For the purpose of Common Core Standards, "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.9).

Step 1: Identify the minimum of the table.

Using the table find the time value where the lowest distance exists.

Recall that the time represents the values while the distance represents the values. Therefore the ordered pair for the minimum can be written as .

Step 2: Identify the minimum of the graph

Recall that the minimum of a parabola opening up, occurs at the valley where the vertex lies.

For this particular graph the vertex is at .

Step 3: Compare the minimums from step 1 and step 2.

Compare the value coordinate from both minimums.

Therefore, the graph has the lowest minimum.

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

For the purpose of Common Core Standards, "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.9).

Step 1: Identify the maximum of Table 1.

Using the table find the time value where the largest distance exists.

Recall that the time represents the values while the distance represents the values. Therefore the ordered pair for the maximum can be written as .

Step 2: Identify the maximum of the graph

Recall that the maximum of a parabola opening down, occurs at the peak where the vertex lies.

For this particular graph the vertex is at .

Step 3: Compare the maximums from step 1 and step 2.

Compare the value coordinate from both maximums.

Therefore, the table has the largest maximum.

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

For the purpose of Common Core Standards, "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.9).

Step 1: Identify the maximum of Table 1.

Using the table find the time value where the largest distance exists.

Recall that the time represents the values while the distance represents the values. Therefore the ordered pair for the maximum can be written as .

Step 2: Identify the maximum of the graph

Recall that the maximum of a cubic function is known as a local maximum. This occurs at the vertex of the peak on the graph which in this particular case, is at the point .

Step 3: Compare the maximums from step 1 and step 2.

Compare the value coordinate from both maximums.

Therefore, the table has the largest maximum.

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

For the purpose of Common Core Standards, "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.9).

Step 1: Identify the minimum of the table.

Using the table find the time value where the lowest distance exists.

Recall that the time represents the values while the distance represents the values. Therefore the ordered pair for the minimum can be written as .

Step 2: Identify the minimum of the graph

Recall that the minimum of a cubic function is known as a local minimum. This occurs at the valley where the vertex lies.

For this particular graph the vertex is at .

Step 3: Compare the minimums from step 1 and step 2.

Compare the value coordinate from both minimums.

Therefore, the graph has the lowest minimum.

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

For the purpose of Common Core Standards, "Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)." falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.9).

Step 1: Identify the maximum of Table 1.

Using the table find the time value where the largest distance exists.

Recall that the time represents the values while the distance represents the values. Therefore the ordered pair for the maximum can be written as .

Step 2: Identify the maximum of the graph

Recall that the maximum of a cubic function is known as a local maximum. This occurs at the vertex of the peak on the graph which in this particular case, is at the point .

Step 3: Compare the maximums from step 1 and step 2.

Compare the value coordinate from both maximums.

Therefore, the table has the largest maximum.