Problem 90 - 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. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 5 & 10 & 15 \\ \hline \boldsymbol{h}(\boldsymbol{x}) & 5 & 30 & 105 & 230 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table does not represent a linear function; differences are not constant.

Step 1: Identify the nature of the relationship

To determine if the table represents a linear function, check if there is a constant rate of change between consecutive outputs. In other words, calculate the difference between consecutive values of \( h(x) \) to see if it is consistent.

Step 2: Calculate the differences

Find the differences between consecutive \( h(x) \) values: \( 30 - 5 = 25 \), \( 105 - 30 = 75 \), and \( 230 - 105 = 125 \). The differences are not constant (25, 75, 125), indicating the relationship is not linear.

Step 3: Determine if it's a linear function

A linear function should have consistent first-level differences. Since the first-level differences (25, 75, 125) vary, this table does not represent a linear function. Hence, a linear equation cannot model this data.

Key Concepts

Linear Equations

A linear equation is a type of mathematical equation that illustrates a straight-line relationship between two variables. This means that if we were to plot the equation on a graph, the result would be a straight line. Linear equations are generally in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The equation suggests that for a given increment in \(x\), \(y\) increases by a constant amount (the slope).

Effectively, linear equations capture situations with a constant rate of change, which is why they show up so often in the study of linear functions.

• Slope \(m\) determines the steepness of the line.
• Y-intercept \(b\) signifies the value of \(y\) when \(x = 0\).

Thus, a linear equation is key in representing uniform relationships across variables. Whether climbing up steps or calculating costs, this framework simplifies many real-world problems for easy understanding and resolution.

Constant Rate of Change

A constant rate of change means that as one variable increases or decreases by a set increment, the other variable changes by a constant amount. This often corresponds to the slope of a linear equation. In simpler terms, in life's daily activities, it’s like having a constant speed as you move from one place to another.

If you notice consistent, uniform differences between values as you increase (or decrease) step-by-step, you're likely observing a constant rate of change. This is indicative in scenarios such as unit pricing, where each additional item costs the same amount, or traveling at a set speed.

• For example, traveling 60 miles every hour demonstrates a constant rate.
• If for every 5 units of \(x\) you increase by, \(y\) always goes up by 10, that’s a constant rate.

Such consistent patterns tell us that the system or function we are looking at can likely be framed using a linear equation. If that rate isn't constant, then the relationship is something other than linear and needs different math tools to explore further.

First-Level Differences

First-level differences are a way to explore if a function is linear by examining changes in output as input increases incrementally. You basically look at how much the dependent variable changes as the independent variable increases by one step. This technique is particularly useful for identifying linear relations in a set of discrete data points.

To identify a linear function using first-level differences:

• Calculate the differences between consecutive outputs.
• Check for consistency.

In a perfect linear function, these differences will be the same across the board, meaning you'll have identified a function with a constant rate of change. If these differences vary, as they did with the table in the original exercise (where we found differences of 25, 75, 125), it tells us the relationship isn't linear.

Understanding first-level differences is a fundamental skill in algebra and can be applied in various fields like economics, physics, and business to deduce straightforward relationships among data points.