Question

describing how to model a relationship with a function

given a table of values, how can you determine whether the function is linear, quadratic, or exponential?

Explanation:

Check for equal x-intervals

First, ensure the input values \(x\) in the table increase by a constant amount (equal intervals, such as \(\Delta x = 1\)). This is a necessary prerequisite before analyzing the patterns in the output values \(y\).

Identify linear relationships

Using the Finite Differences and Polynomial Model Selection knowledge points

• Calculate the first differences of consecutive \(y\)-values: \(y_{n} - y_{n-1}\).
• If these first differences are constant, the relationship is linear.

Identify quadratic relationships

Using the Finite Differences and Polynomial Model Selection knowledge points

• Calculate the second differences by finding the differences of the first differences.
• If these second differences are constant, the relationship is quadratic.

Identify exponential relationships

Using the Exponential Modeling knowledge point

• Calculate the ratios of consecutive \(y\)-values: \(\frac{y_{n}}{y_{n-1}}\).
• If these ratios are constant, the relationship is exponential.

Answer:

To determine whether a function represented by a table of values is linear, quadratic, or exponential, first ensure that the \(x\)-values increase by a constant interval. Then, analyze the \(y\)-values:

1. Linear: The first differences between consecutive \(y\)-values are constant.
2. Quadratic: The second differences (the differences of the first differences) are constant.
3. Exponential: The ratios of consecutive \(y\)-values are constant.