Question

Determine whether each function is linear or nonlinear. If it is linear, determine the slope. $$ \begin{array}{|rr|} \hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\ \hline-2 & -8 \\ -1 & -3 \\ 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ \hline \end{array} $$

Short answer

The function is nonlinear.

Step-by-step solution

- Identify Function Type

Write down the given points and inspect the general behavior. The points are (-2, -8), (-1, -3), (0, 0), (1, 1), and (2, 0). Plot these points on a graph to analyze whether the data forms a straight line or not.

- Check for Constant Slope

Calculate the slope between each pair of points to see if it remains constant. Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Calculate the slope between (-2, -8) and (-1, -3), between (-1, -3) and (0, 0), between (0, 0) and (1, 1), and between (1, 1) and (2, 0).

- Slope Calculations

Compute the slopes:\[ m_1 = \frac{-3 - (-8)}{-1 - (-2)} = \frac{5}{1} = 5 \]\[ m_2 = \frac{0 - (-3)}{0 - (-1)} = \frac{3}{1} = 3 \]\[ m_3 = \frac{1 - 0}{1 - 0} = 1 \]\[ m_4 = \frac{0 - 1}{2 - 1} = \frac{-1}{1} = -1 \] The slopes are different which confirms that the function is not linear.

- Conclusion

Since the slopes between the points are not constant, the function represented by the given points is nonlinear.

Key concepts

Slope Calculation

Understanding how to calculate the slope is essential for determining whether a function is linear or nonlinear. The slope of a line measures its steepness and is often represented by the letter \( m \). To find the slope between two points, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula essentially takes the difference in the \( y \)-values and divides it by the difference in the \( x \)-values. If the slope is constant (the same between any pair of points), then the function is linear.
In our example, the points given were: (-2, -8), (-1, -3), (0, 0), (1, 1), and (2, 0). We used the slope formula to calculate the slopes between each adjacent pair of points:

• From (-2, -8) to (-1, -3): \( m = 5 \)
• From (-1, -3) to (0, 0): \( m = 3 \)
• From (0, 0) to (1, 1): \( m = 1 \)
• From (1, 1) to (2, 0): \( m = -1 \)

Since these slopes are different, we conclude that the function is nonlinear.

Graph Analysis

Graphing points is a very effective way to visually determine if a function is linear or nonlinear. When you plot the points given in a function, if they all fall on a straight line, then the function is linear. If they do not all form a single straight line, the function is nonlinear.
For our exercise:

• The points plotted were (-2, -8), (-1, -3), (0, 0), (1, 1), and (2, 0).
• On graphing, you'll notice that these points do not align perfectly along a single straight line.
• Therefore, the graph does not support that the function is linear.

This visual inspection, along with slope calculation, helps confirm whether a function is nonlinear. When analyzing graphs, always plot your points and check for linearity by observing the alignment of the points.

Nonlinear Functions

Nonlinear functions are those that do not form a straight line when graphed. Unlike linear functions, the rate of change in nonlinear functions is not constant. This means that the slope between different pairs of points will vary.
Characteristics of nonlinear functions include:

• Variable rate of change.
• Graph that could be curved, triangular, parabolic, or any shape other than a straight line.
• Slope calculations that yield different values between different pairs of points.

In our example, we found the function was nonlinear as the slopes calculated between each pair of points were different. A nonlinear function is identified not just by the uneven slopes but also by the shape it forms on a graph.
Understanding these patterns can help in identifying and analyzing nonlinear functions in a variety of mathematical contexts.