Problem 96 - 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. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 2 & 6 & 8 \\ \hline \boldsymbol{k}(\boldsymbol{x}) & 6 & 31 & 106 & 231 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table cannot represent a linear function as the slopes are not constant.

Step 1: Examine Table for Linearity

To determine if the table represents a linear function, we need to check if the changes in the output values, as the input values change, are constant. This is done by checking if the difference between consecutive output values divided by the difference in consecutive input values (slope) is constant.

Step 2: Calculate Slopes Between Points

Calculate the slope between each pair of consecutive points in the table:1. Between (0, 6) and (2, 31), slope = \( \frac{31-6}{2-0} = \frac{25}{2} = 12.5 \)2. Between (2, 31) and (6, 106), slope = \( \frac{106-31}{6-2} = \frac{75}{4} = 18.75 \)3. Between (6, 106) and (8, 231), slope = \( \frac{231-106}{8-6} = \frac{125}{2} = 62.5 \)Each computed slope is different, indicating that the relationship is not constant.

Step 3: Conclude Non-linearity

Since the slopes between consecutive points are not equal, the function represented by this table is not linear.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is essential for analyzing linear functions. The slope tells us the rate of change between two points on a line. It's represented by the ratio of the difference in the y-values to the difference in the x-values. In our exercise, we calculate the slope between points in a table. To find the slope, use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This expresses the change in the function values (outputs) per unit change in input values. Each pair of consecutive points yields a slope value. If these values differ, the function is not linear.In the example, we calculated three different slopes: 12.5, 18.75, and 62.5. None of them are the same, which means the changes are not consistent, indicating non-linearity. Slope calculations help determine if there is uniform growth or decline in data presented in tables or graphs.

Linearity Test

A linearity test is a useful method to decide if a function is linear by examining if the graph or table data forms a straight line. This is done using the slopes between consecutive data points. If these slopes are equal, the function can be considered linear. The test involves checking for a constant slope across the entire dataset:

• Find slopes of each pair of consecutive points.
• Compare these slopes.
• If all are the same, the data describes a linear function.

In our case, by calculating slopes between each pair of data points and finding that they differ, we concluded the function isn't linear. A consistent slope is an indicator of linearity, where each horizontal step corresponds to a consistent vertical rise or fall.

Non-linear Function

When analyzing functions, determining non-linearity is equally crucial. Non-linear functions do not form a straight line when graphically represented. Their slopes vary between different points, leading to curves or uneven changes in the output values. In the example exercise, since each calculated slope is distinct, this implies that the relationship between inputs and outputs cannot be captured by a single straight line. Key characteristics of non-linear functions include:

• Changing slopes between points.
• Curved or wiggly graphical representations.
• Potential exponents, squares, or other operations in equations.

Recognizing non-linearity helps determine that multiple factors or variations affect the relationship within the data, rather than a single, constant rate of change.