Question

Determine whether each function is linear or nonlinear. If it is linear, determine the slope. $$ \begin{array}{|rc|} \hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\ \hline-2 & 4 \\ -1 & 1 \\ 0 & -2 \\ 1 & -5 \\ 2 & -8 \\ \hline \end{array} $$

Short answer

The function is linear with a slope of -3.

Step-by-step solution

- Understand the Problem

The given function is represented by a set of points. Determine whether it is linear or nonlinear and then find the slope if it is linear.

- Check for Linearity

To check if the function is linear, determine if the rate of change between each consecutive pair of points is constant. Calculate the rate of change (slope) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

- Calculate Slopes

Calculate the slopes between consecutive points: \[ m_{1} = \frac{1 - 4}{-1 - (-2)} = \frac{-3}{1} = -3 \] \[ m_{2} = \frac{-2 - 1}{0 - (-1)} = \frac{-3}{1} = -3 \] \[ m_{3} = \frac{-5 - (-2)}{1 - 0} = \frac{-3}{1} = -3 \] \[ m_{4} = \frac{-8 - (-5)}{2 - 1} = \frac{-3}{1} = -3 \]

- Analyze

Since the slope between each pair of consecutive points is the same, the function is linear.

- State the Result

The function is linear with a constant slope of \( -3 \).

Key concepts

Determine Linearity

Determining if a function is linear or non-linear is essential. A function is linear if its rate of change (or slope) remains consistent between all its points. Linear functions graph to a straight line, while non-linear functions do not.

To determine linearity: - Calculate the slopes (or rates of change) between each consecutive set of points using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1} \) - Check if these slopes are equal.

In the provided solution, the function was confirmed linear as every calculated slope was 'm'= -3. If the slope varied, the function would be non-linear.