Problem 89 - College Algebra

Question

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 5 & 10 & 15 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 5 & -10 & -25 & -40 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Yes, the function is linear; equation: \(g(x) = -3x + 5\).

Step 1: Understanding a Linear Function

A linear function has a constant rate of change or slope. This means that for equal intervals in the x-values, the change in y-values must be the same. We will check the table values to see if they meet this criterion.

Step 2: Calculate the Difference in x-values

First, find the difference between successive x-values to ensure that they are consistent. Given x-values are: \(0, 5, 10, 15\). The difference between each pair is 5 since \(5 - 0 = 5\), \(10 - 5 = 5\), and \(15 - 10 = 5\). Thus, the difference in x-values is consistent.

Step 3: Calculate the Difference in g(x)-values

Next, find the difference between successive \(g(x)\)-values: \(5, -10, -25, -40\). The differences are: \(-10 - 5 = -15\), \(-25 - (-10) = -15\), and \(-40 - (-25) = -15\). The change in \(g(x)\)-values is constant across the intervals suggesting a linear function.

Step 4: Determine the Slope

The slope \(m\) of a linear function is the change in y-values over the change in x-values. Using the differences we found, the slope \(m\) is \(-15/5 = -3\).

Step 5: Find the y-intercept

A linear equation has the form \(y = mx + b\). To find \(b\), the y-intercept, use one of the points in the table. Let's use \((0, 5)\). Substitute \(x = 0\) and \(g(x) = 5\) into the equation: \[5 = -3(0) + b\],thus \(b = 5\).

Step 6: Write the Linear Equation

Now that we have both the slope \(m = -3\) and the y-intercept \(b = 5\), we can write the linear equation as: \[g(x) = -3x + 5\].

Key Concepts

Slope Calculation

When determining if a function is linear, the first step is to calculate the slope. Slope is a measure of how much one quantity changes in relation to another. In a linear function, the slope, denoted as \( m \), is constant. This means that for equal increases in \( x \), the changes in \( g(x) \) remain constant at every interval.
To calculate the slope, you divide the difference in the \( y \)-values by the difference in the \( x \)-values. For instance, if the values change from \(-10\) to \(5\), the slope \( m \) would be calculated as:

• Calculate the difference in \( g(x) \)-values: \(-10 - 5 = -15\)
• Calculate the difference in \( x \)-values: \(5 - 0 = 5\)
• Finally, compute the slope: \( m = \frac{-15}{5} = -3 \)

This consistent rate of change indicates how steep the line is and in what direction it goes.

Linear Equation

A linear equation describes a straight line in algebra. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This equation tells us how \( y \) changes for every unit increase in \( x \).
Understanding the linear equation involves knowing three key parts:

• The slope \( m \), which determines the tilt of the line
• The \( x \) variable, which represents the independent variable
• The \( b \) or y-intercept, showing where the line crosses the y-axis

Once you've determined \( m \) and \( b \), you can express the linear function clearly. For example, if \( m = -3 \) and \( b = 5 \), the equation becomes \( g(x) = -3x + 5 \). This equation gives the complete picture of how your linear function behaves.

Rate of Change

The rate of change in a linear function is crucial to understand. It signifies how much the dependent variable changes with respect to a change in the independent one. For linear functions, this is simply the slope. The rate of change tells us if a function is increasing or decreasing and how quickly.
An increasing function has a positive rate of change, and the line tilts upwards. Conversely, a negative rate of change means the function decreases, tilting downwards.
In our example, the rate of change is \(-3\). For every increase of 1 in \( x \), \( g(x) \) decreases by 3. This showcases a constant downward trend.

Y-Intercept

The y-intercept \( b \) is the point where the graph of the linear function crosses the y-axis. Physically, it can often represent the start or initial value of a function when the independent variable \( x \) is zero.
To find the y-intercept in our equation, recognize when \( x = 0 \). In the equation \( g(x) = -3x + 5 \), setting \( x \) to zero leaves us with \( g(0) = 5 \). Thus, the y-intercept is 5.
This means that no matter what the slope is, the graph will always start from this point on the y-axis. Visually, it sets the initial height of the line on a graph.