91Ó°ÊÓ

This exercise shows that if we have a table generated by a linear function and the \(x\) -values are equally spaced, then the first differences of the \(y\) -values are constant. (a) In the following data table, the three \(x\) -entries are equally spaced. Compute the three entries in the \(f(x)\) row assuming that \(f\) is the linear function given by \(f(x)=m x+b\) \(f(x)=m x+b .\) (Don't worry about the fact that your answers contain all four of the letters \(m, b, a, \text { and } h .)\) \begin{tabular}{llll} \hline\(x\) & \(a\) & \(a+h\) & \(a+2 h\) \\ \(f(x)\) & & & \\ \hline \end{tabular} (b) Compute the first differences for the three quantities that you listed in the \(f(x)\) row in part (a). (The two first differences that you obtain should turn out to be equal, as required.)

Step by step solution

01

Fill the Table Using the Linear Function

The function given is \(f(x) = mx + b\). We are asked to fill the corresponding values in the \(f(x)\) row for the given \(x\)-values \(a\), \(a+h\), and \(a+2h\). We do this by substituting each \(x\)-value into the function:- For \(x = a\), \(f(a) = ma + b\)- For \(x = a + h\), \(f(a + h) = m(a + h) + b = ma + mh + b\)- For \(x = a + 2h\), \(f(a + 2h) = m(a + 2h) + b = ma + 2mh + b\)The completed table will be:\[\begin{tabular}{|c|c|c|c|}\hlinex & a & a+h & a+2h \\hlinef(x) & ma+b & ma+mh+b & ma+2mh+b \\hline\end{tabular}\]

02

Compute the First Differences of f(x) Values

The first difference is obtained by subtracting the consecutive \(f(x)\) values:- First difference between \(f(a+h)\) and \(f(a)\): \[f(a+h) - f(a) = (ma + mh + b) - (ma + b) = mh\]- First difference between \(f(a+2h)\) and \(f(a+h)\): \[f(a+2h) - f(a+h) = (ma + 2mh + b) - (ma + mh + b) = mh\]Both first differences are equal to \(mh\), confirming that they are constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Differences

Understanding **first differences** is crucial when analyzing linear functions. First differences help determine if a function has a constant rate of change, which is one characteristic of linear functions. They are essentially the differences of consecutive terms in a sequence of values from a function. For a function of the form \(f(x) = mx + b\), first differences are found by subtracting the value of the function at consecutive points.

• The first difference between \(f(a + h)\) and \(f(a)\) is derived by \(f(a + h) - f(a)\).
• Similarly, the first difference between \(f(a + 2h)\) and \(f(a + h)\) is \(f(a + 2h) - f(a + h)\).

For linear functions, these first differences remain constant, demonstrating that the function increases or decreases at a steady rate.
This concept is fundamental to identifying linear relationships in the data effectively.

Equally Spaced Data

When we talk about **equally spaced data**, we mean that the input values \(x\) for the function are separated by the same interval or difference. In the exercise, the \(x\)-values \([a, a+h, a+2h]\) are evenly spaced, with each step of \(h\).Using equally spaced intervals in linear functions simplifies the computation of first differences and helps identify a constant rate of change. Additionally:

• This spacing allows us to more easily verify the linearity of a function by illustrating the constant first differences.
• Consistent intervals ensure that corresponding changes in the \(y\)-values align directly with the linear function's rate of change.

Equal spacing is a simple yet vital attribute when dealing with linear functions as it maintains uniformity, making it easier to predict further values and understand the behavior of the function.

Constant Rate of Change

A **constant rate of change** is a fundamental characteristic of linear functions, indicating that the function increases or decreases by the same amount over each step of \(x\).The slope \(m\) in the linear equation \(f(x) = mx + b\) represents this constant rate:

• If the slope \(m\) is positive, the function rises consistently, and if negative, it falls consistently.
• In the exercise, we see this demonstrated through equal first differences of \(mh\), proving the change from one step to the next remains constant.

This consistency allows for easier predictions and calculations, as it tells us that the same difference in one variable results in the same difference in its function counterpart. Understanding constant rate of change is crucial for modeling real-world phenomena with linear functions, providing a straightforward approach to analyzing changes and trends. In simple terms, this constant behavior is what links the mathematics with practical applications.