Question

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

Short answer

The function is linear with a slope of zero.

Step-by-step solution

Identify the Definition of a Linear Function

A function is linear if it can be written in the form of a linear equation: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Analyze the Given Data

Examine the given points to determine if they form a straight line when plotted on a graph. The points provided are \((-2, 8)\), \((-1, 8)\), \((0, 8)\), \((1, 8)\), and \((2, 8)\).

Determine if the Function is Linear

Notice that for each value of \(x\), the value of \(y\) remains constant at 8. This suggests that the function does not change with \(x\), indicating it is a horizontal line.

Write the Function Equation

Since \(y\) is always 8, the function can be written as \(y = 8\). This is a linear function because it fits the form \(y = mx + b\) where \(m = 0\) (no slope) and \(b = 8\).

Determine the Slope

The slope \(m\) of the function is 0 since the value of \(y\) does not change as \(x\) changes. A horizontal line has a slope of zero.

Key concepts

slope

The slope of a linear function measures how steep the line is. Mathematically, the slope is represented by the letter 'm' in the linear equation form, \(y = mx + b\). The slope is calculated as the ratio of the change in the y-values to the change in the x-values between two points on the line. It can be found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

In the given example, we have the points \((-2, 8)\), \((-1, 8)\), \((0, 8)\), \((1, 8)\), and \((2, 8)\). Notice that the y-value (8) is the same for all points, meaning there is no vertical change as x changes. Thus, the slope 'm' is calculated as:

\[ m = \frac{8 - 8}{x_2 - x_1} = 0 \]
This means the line is perfectly horizontal, having a slope of zero. A zero slope indicates no uphill or downhill inclination, remaining completely flat.

linear equation

A linear equation represents a straight line on a graph and can be written in the form \(y = mx + b\). Here:

• 'm' is the slope of the line.
• 'b' is the y-intercept, the point where the line crosses the y-axis.

A function is considered linear if it can be expressed in this form and if the graph of the equations shows a straight line.

In the original exercise, we're given several points and need to ascertain if they form a linear function. The data presents itself as follows:

• \((-2, 8)\)
• \((-1, 8)\)
• \((0, 8)\)
• \((1, 8)\)
• \((2, 8)\)

Since the y-values are the same for any x-value, we can see that there is no change in the y-axis; thus, it's a horizontal line. The linear equation for this set of data is:

\[y = 0x + 8 = 8\]
In this case, the slope (m) is 0, and the y-intercept (b) is 8.

horizontal line

A horizontal line is a special type of linear equation. It shows no change in the y-values, meaning the slope is zero. On a graph, a horizontal line runs left-to-right and remains at the same y-value, regardless of the x-values.

In the given example, the line can be written as \(y = 8\), indicating that for any x-value, y will always be 8. Characteristics of a horizontal line include:

• Slope (m) of 0, making the line flat.
• Equation in the form of \(y = k\), where 'k' is the constant y-value.
• No vertical movement or inclination.

Identifying horizontal lines is simple. As long as the y-value does not change regardless of fluctuations in the x-values, the line remains horizontal. The given points in the exercise demonstrate this perfectly, so the solution confirms the line \(y = 8\) is linear and horizontal.