The function in the table below shows the relationship between the total number of houses built in an area and the number of months that passed.

Question

A two column table with five rows. The first column, Months Passed, has the entries, 0, 3, 4, 8. The second column, Total Houses Built, has the entries 0, 33, 46, 108.

Which best describes the data set?

It is nonlinear because the “Total Houses Built” column does not increase at a constant additive rate.
It is nonlinear because the “Months Passed” column does not increase at a constant additive rate.
It is nonlinear because the increase in the “Total Houses Built” compared to the increase in the “Months Passed” does not show a constant rate of change.
It is linear because the increase in the “Total Houses Built” compared to the increase in the “Months Passed” shows a constant rate of change

Answer 1

To determine whether the data set is linear or nonlinear, we need to examine the changes in the "Total Houses Built" with respect to changes in the "Months Passed."

Let's break down the data:

• From month 0 to month 3:

  • Houses built = 33 - 0 = 33
  • Months passed = 3 - 0 = 3
  • Rate of change = 33 houses / 3 months = 11 houses per month

• From month 3 to month 4:

  • Houses built = 46 - 33 = 13
  • Months passed = 4 - 3 = 1
  • Rate of change = 13 houses / 1 month = 13 houses per month

• From month 4 to month 8:

  • Houses built = 108 - 46 = 62
  • Months passed = 8 - 4 = 4
  • Rate of change = 62 houses / 4 months = 15.5 houses per month

Now we can summarize the rates of change:

• From 0 to 3 months: 11 houses per month
• From 3 to 4 months: 13 houses per month
• From 4 to 8 months: 15.5 houses per month

Since the rates of change are not constant (11, 13, and 15.5), this indicates that the relationship is not linear.

Therefore, the correct statement that best describes the data set is:

It is nonlinear because the increase in the “Total Houses Built” compared to the increase in the “Months Passed” does not show a constant rate of change.

Answer 2

A coordinate plane showing Nina's run. The x-axis shows Time in seconds and the y-axis shows Distance in meters. Four points plotted and labeled. The points are (4, 32), (6, 48), (8, 64), (10, 80). A two column table with four rows. The first column, Time in seconds, has the entries, 4, 6, 8. The second column, Distance in meters, has the entries, 35, 47.5, 60.
Nina and Ryan each ran at a constant speed for a 100-meter race. Each runner’s distance for the same section of the race is displayed on the left. Who had a head start, and how big was the head start?____

had a head start of
meters.____

Answer 3

To determine who had a head start and how large it was, we need to compare Nina's distance to Ryan's distance at the same time intervals.

Nina's Distance (from the first set) at Time in Seconds:

• At 4 seconds: 32 meters
• At 6 seconds: 48 meters
• At 8 seconds: 64 meters
• At 10 seconds: 80 meters

Ryan's Distance (from the table):

• At 4 seconds: 35 meters
• At 6 seconds: 47.5 meters
• At 8 seconds: 60 meters

Comparison at 4 seconds:

• Nina: 32 meters
• Ryan: 35 meters

At 4 seconds, Ryan is ahead with 35 meters while Nina has only run 32 meters. Ryan is ahead by:

• Head start = Ryan's distance - Nina's distance = 35 - 32 = 3 meters.

Comparison at 6 seconds:

• Nina: 48 meters
• Ryan: 47.5 meters

At 6 seconds, Nina exceeded Ryan's distance by:

• Head start = Nina's distance - Ryan's distance = 48 - 47.5 = 0.5 meters.

Comparison at 8 seconds:

• Nina: 64 meters
• Ryan: 60 meters

At 8 seconds, Nina is ahead:

• Head start = Nina's distance - Ryan's distance = 64 - 60 = 4 meters.

Conclusion

Initially, Ryan had a head start of 3 meters because he was ahead of Nina at 4 seconds. As the race progressed, Nina eventually surpassed Ryan after that point.

So, the final answer: Ryan had a head start of 3 meters.

Answer 4

A coordinate plane showing Nina's run. The x-axis shows Time in seconds and the y-axis shows Distance in meters. Four points plotted and labeled. The points are (4, 32), (6, 48), (8, 64), (10, 80). A two column table with four rows. The first column, Time in seconds, has the entries, 4, 6, 8. The second column, Distance in meters, has the entries, 35, 47.5, 60.
Nina and Ryan each ran at a constant speed for a 100-meter race. Each runner’s distance for the same section of the race is displayed on the left. Who had a head start, and how big was the head start?

______had a head start of
meters.________

Answer 5

To determine who had a head start and how large it was, we need to compare their distances at the same time intervals.

Nina's Distance:

• At 4 seconds: 32 meters
• At 6 seconds: 48 meters
• At 8 seconds: 64 meters
• At 10 seconds: 80 meters

Ryan's Distance (from the table):

• At 4 seconds: 35 meters
• At 6 seconds: 47.5 meters
• At 8 seconds: 60 meters

Now, let's compare their distances at the time intervals given:

At 4 seconds:

• Nina: 32 meters
• Ryan: 35 meters

Ryan is ahead by:

• \(35 - 32 = 3\) meters

At 6 seconds:

• Nina: 48 meters
• Ryan: 47.5 meters

Nina is ahead by:

• \(48 - 47.5 = 0.5\) meters

At 8 seconds:

• Nina: 64 meters
• Ryan: 60 meters

Nina is ahead by:

• \(64 - 60 = 4\) meters

Conclusion

Initially, at 4 seconds, Ryan is ahead by 3 meters. Therefore, we conclude:

Ryan had a head start of 3 meters.

Answer 6

Ryan with 10 meters

What is the difference between a linear function and a nonlinear function? Explain what each looks like when represented as a table and as a graph.

Answer 7

A linear function and a nonlinear function differ fundamentally in their rate of change, behavior, and representation.

Linear Function

Definition:

A linear function is a function whose graph is a straight line. It can be expressed in the standard form \(y = mx + b\), where:

• \(m\) is the slope (rate of change).
• \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).

Characteristics:

• The rate of change (slope) between any two points is constant.
• The equation of a linear function represents a proportional relationship among the variables.

Table Representation:

In a table representing a linear function, if you observe the values for \(y\) as \(x\) changes, the differences between successive \(y\) values will remain consistent.

| \(x\) | \(y\) | |-------|-------| | 0 | 2 | | 1 | 4 | | 2 | 6 | | 3 | 8 | | 4 | 10 |

In this table:

• The difference in \(y\) (output) is constant as \(x\) (input) increases.

Graph Representation:

The graph of a linear function is a straight line. Each point on the line represents a solution to the equation of the function.

Nonlinear Function

Definition:

A nonlinear function is one in which the relationship between the independent variable and the dependent variable is not a straight line. It can take various forms, such as quadratic, exponential, or logarithmic functions.

Characteristics:

• The rate of change is not constant; it can accelerate or decelerate.
• The equation may not be expressible in the linear format \(y = mx + b\).

Table Representation:

In a table for a nonlinear function, the differences between successive \(y\) values may change.

| \(x\) | \(y\) | |-------|-------| | 0 | 1 | | 1 | 2 | | 2 | 4 | | 3 | 9 | | 4 | 16 |

Here, you can see that the differences in \(y\) values (1, 2, 5, 7) are not constant, indicating a nonlinear relationship.

Graph Representation:

The graph of a nonlinear function might be a curve, such as a parabola or an exponential curve.

In summary:

• Linear functions produce straight lines with constant rates of change, and their tables show consistent differences in \(y\).

• Nonlinear functions produce curves and have variable rates of change, with their tables showing varying differences in \(y\) values as \(x\) changes.