How to Write Linear Functions from Tables

A linear function is a polynomial function of degree one, presenting a straight line when graphed. The general format of a linear function is \(f(x)=mx+b\), where \(m\) is the slope, and \(b\) is the \(y\)-intercept.

Tables are a succinct way to represent data. When data points from a linear function are presented in a table, it’s essential to be adept at interpreting these tables to decipher the function.

A Step-by-step Guide to Writing Linear Functions from Tables

Here is a step-by-step guide on how to write linear functions from tables:

Step 1: Identify the Variables

The first step in writing a linear function from a table is identifying the variables. Typically, tables for linear functions feature two columns, one for each variable (\(x\) and \(y\)).

Step 2: Determine the Slope

The slope (\(m\)) of a line is the rate at which \(y\) changes for each change in \(x\). To find the slope, subtract the \(y\)-value of the second row from the \(y\)-value of the first row. Then subtract the x-value of the second row from the \(x\)-value of the first row. Dividing these two differences gives you the slope: \(m =\frac{y2 – y1}{x2 – x1}\).

Step 3: Calculate the Y-Intercept

The y-intercept (\(b\)) is the point at which the line crosses the \(y\)-axis. This point can be directly seen in the table as the \(y\)-value when \(x\) equals zero. If the \(x = 0\) value isn’t available in the table, you can use the slope (\(m\)) and one set of coordinates \((x, y)\) from the table to solve for \(b\) using the formula: \(b=y-mx\).

Example: Creating a Linear Function from a Table

Consider the following table:

To determine the slope, subtract the \(y\)-values \((5-3=2)\) and the \(x\)-values \((2-1=1)\) between the first two rows. The slope \(m=\frac{2}{1}=2\).

For the y-intercept, you can see when \(x =1, y=3\). Using the slope (\(2)\), and substituting \(x (1)\) and \(y (3)\) into the formula \(b=y-mx\), we find that \(b=3-2\times 1 = 1\).

Thus, the linear function corresponding to this table is \(f(x)=2x+1\) or simply, \(f(x)=2x+1\).

The Real-World Relevance of Linear Functions

Understanding how to write linear functions from tables is not just critical in mathematics but also in real-world scenarios. These scenarios include predicting sales growth, estimating profit margins, and even modeling natural phenomena such as population growth.