Linear: y=ax + b (Junior Cert Mathematics): Revision Notes

Linear Functions

What is a Linear Function?

A linear function is a type of mathematical function that creates a straight line when graphed on a coordinate plane. The standard form of a linear function is:

$f(x) = mx + c$

Where:

• $f(x)$is the output or the y$$-value (sometimes written as $y$).
• $x$ is the input or the x$$-value.
• $m$ is the slope of the line, which tells us how steep the line is.
• $c$ is the y-intercept, which is the point where the line crosses the $y-axis$.

Why Are They Called Linear Functions?

Linear functions are called "linear" because they graph as straight lines. To plot a function on a graph, we place the input values on the x$$-axis (horizontal axis) and the corresponding output values on the y$$-axis (vertical axis). When you plot different points of the function on a graph and then connect them, you will get a straight line.

Plotting a Linear Function

To plot a linear function on a graph, follow these steps:

1. Choose a value for $x$ (this is your input value). It's a good idea to pick a few different values to get a more accurate graph.
2. Calculate the corresponding $y$ value using the function (this is your output value).
3. Plot the points on the graph. Remember that the x $$-value goes on the horizontal axis (x$$-axis) and the y $$-value goes on the vertical axis (y$$-axis).
4. Connect the points with a straight line. This line represents the linear function.

Once you've plotted these points, draw a straight line through them to represent the function. The line extends infinitely in both directions, showing all possible points for the function.

Summary

• A linear function creates a straight line when graphed.
• The function can be written as $y = mx + c$.
• You can graph a linear function by plotting points.
• The x$$-axis represents the input values, and the y$$-axis represents the output values.