A linear function, as it is called, is an algebraic expression that represents the graph of a straight line.

When we talk about functions, it's important to highlight that the graphs of functions are represented in an axis system where there is a horizontal axis $X$ and a vertical axis $Y$.

Linear functions can be expressed by the expressions $y = mx$ or $y = mx + b$, where m represents the slope of the line while $b$ (when it exists) represents the y-intercept.

To plot a linear function, all we need are $2$ points. If the linear function is given, you can substitute a value for $X$ and obtain the corresponding $Y$ value.

We are asked to graph it on the coordinate system.

As we have discussed, to do this we need two points, which we will place in the function's expression. Choose any two points we like, it doesn't matter.

Now we will plot the two points on the coordinate system and connect them. This is actually a graph of the function for $y=2x+1$.

Examples and Exercises with Solutions for Linear Functions

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Find the slope of the line that passes through the points $(4,1),(2,5)$

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Remember the formula to calculate the slope using the points:

Now, replace the data in the formula:

$\frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2$

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Test your knowledge

Question 1

Below is a table containing values for x and y. This tables represents a linear function.

Choose the equation that corresponds to the function.