# How to Write Linear Functions from Tables

The concept of linear functions is one of the key topics in algebra and fundamental to understanding the world of mathematics. 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.

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