Describe a situation that models a linear pattern and then describe a situation that models a nonlinear pattern. Do not state which situation is linear and which is nonlinear. Your classmates will need to determine which is which and then write a function that models the linear situation. Your classmates will also need to write 1 ordered pair that is a solution to the linear function and explain its meaning in the context of the situation.
Linear Situation:
The situation involves a car driving on a straight road at a constant speed. As the car travels, the distance it covers is directly proportional to the time it takes. For example, if the car drives at a constant speed of 60 miles per hour, then after 1 hour it will have traveled 60 miles, after 2 hours it will have traveled 120 miles, and so on.
Nonlinear Situation:
The situation involves a plant growing in a garden. As the days pass, the plant's height increases, but not at a constant rate. In the beginning, the plant growth may be slow, but as time goes on, it starts to grow faster. For instance, after 1 day, the plant may have grown 1 inch, after 2 days it may have grown 3 inches, after 3 days it may have grown 6 inches, and so on.
It is important to note that the first situation involving the car driving on a straight road represents a linear pattern, while the second situation involving the plant's growth represents a nonlinear pattern.
Situation 1:
A student is studying for a test and their score increases every hour they study. For example, after studying for 1 hour, they score 75. After studying for 2 hours, they score 85. After studying for 3 hours, they score 95. This pattern continues consistently.
Situation 2:
A plant is growing in a garden. Initially, the growth of the plant is slow. However, as time goes on, the plant starts growing faster and faster. For example, after 1 week, the plant has grown 2 inches. After 2 weeks, the plant has grown 6 inches. After 3 weeks, the plant has grown 12 inches. The rate of growth increases irregularly over time.
In the linear situation, the function that models the pattern can be written as y = mx + c, where x represents the number of hours studied and y represents the corresponding score. To find the values of m and c, we can select two points from the given examples and solve for the variables.
For example, using the points (1, 75) and (2, 85):
Using the point-slope form, m = (85 - 75) / (2 - 1) = 10
Using the slope-intercept form, c = 75 - 10(1) = 65
Therefore, the linear function that models the situation is y = 10x + 65.
One ordered pair that would be a solution to this linear function, for example, is (4, 105). This means that after studying for 4 hours, the student is expected to score 105.
In the context of the situation, the ordered pair (4, 105) represents the score the student is predicted to achieve after studying for 4 hours.
Sure, here are two situations that could model either a linear or nonlinear pattern:
Situation 1: A company is producing widgets. Every day, they produce 10 more widgets than the previous day.
Situation 2: A math class is learning about exponential growth. Each week, the number of students who understand the concept doubles.
To determine which situation models a linear pattern and which models a nonlinear pattern, we need to analyze the rate of change between values. In Situation 1, the rate of change is constant (+10 widgets/day), indicating a linear pattern. In Situation 2, the rate of change is not constant; instead, it is increasing exponentially, suggesting a nonlinear pattern.
Now, let's focus on the linear situation (Situation 1) and write a function that models it. We can define the number of widgets produced each day as "W" and the number of days as "D". The function would be:
W = 10D
In this function, "D" represents the number of days since the company started production, and "W" represents the corresponding number of widgets produced. For example, if the company has been producing widgets for 5 days, we can plug in D = 5:
W = 10 * 5
W = 50
Therefore, when D = 5, W = 50, meaning that after 5 days of production, the company has produced 50 widgets.
The ordered pair (5, 50) represents the solution to the linear function and holds meaning within the context of the situation. In this case, it signifies that after 5 days of production, the company has produced a total of 50 widgets.
Remember, it is now up to your classmates to determine whether Situation 1 (linear) or Situation 2 (nonlinear) matches the descriptions above.