In this worksheet, the task is to determine whether each scenario is best represented by a linear, quadratic, or exponential relationship. You also need to give a reason for each type of relation based on the given data or equation.

Let's analyze each example:

1. The table showing the cost of a pizza based on the number of toppings:

• The data in the table is as follows:
  • 0 toppings → $10
  • 1 topping → $12
  • 2 toppings → $14
  • 3 toppings → $16
• Type of Relation: This is a linear relationship because the cost increases by a constant amount ($2) with each additional topping.
• Reasoning: The difference between successive values is constant, which is a key characteristic of linear relationships.

2. A company's profit described by the equation $P = -0.5n^2 - 5n - 1000$:

• Type of Relation: This is a quadratic relationship.
• Reasoning: The equation is in the form of a quadratic function $P = an^2 + bn + c$, where the highest power of $n$ is 2. Quadratic equations often model scenarios where there is a maximum or minimum point, such as profits increasing to a peak and then decreasing.

3. A colony of 4 bacteria doubling every hour (described by the equation $P = 4(2^t)$):

• Type of Relation: This is an exponential relationship.
• Reasoning: The population is doubling, and the equation has the form $P = ab^t$, where $b > 1$ indicates exponential growth. In this case, the population doubles, which is typical of exponential growth models.

4. Graph showing the decay of a radioactive substance:

• Type of Relation: This is an exponential decay.
• Reasoning: The graph shows a steep decline that slows down over time, which is typical of exponential decay. The percentage of remaining radioactive substance decreases rapidly at first and then more slowly, a characteristic feature of exponential decay.

Do you want further details on any of these points?

Here are 5 relative questions to expand understanding:

1. How can you tell if a data set represents a quadratic relation from a table?
2. What are other real-life examples of exponential growth and decay?
3. Can a relationship be both quadratic and linear under certain conditions?
4. What is the difference between an exponential growth graph and an exponential decay graph?
5. How do linear relationships differ from quadratic in terms of graph shapes?

Tip: Always check if differences between consecutive values are constant to quickly identify a linear relationship from a table.