See also General Function Explorer where you can graph up to three functions of your choice simultaneously using sliders for independent variables as above.
Linear functions are those where the independent variable x never has an exponent larger than 1. So for example they would not have a var such as 3x^{2} in them. The linear function on this page is the general way we write the equation of a straight line. It is of the form
y = ax + b 
Where:

The a var is the slope of the line and controls its 'steepness'. A positive value has the slope going up to the right. A negative slope goes down to the right.
The b var is the y intercept  the point where the line crosses the y axis. Adjust the sliders above to vary the values of a and b, and note the effects they have on the graph.
The more common form of the linear function is written y = mx+b, using m for the slope instead of a. This version is included to be consistent with the quadratic and cubic explorers. If you prefer it the usual way use Linear explorer (mx+b).
Since a and b are both set to zero, this is the graph of the equation y = 0x+0. This simplifies to y = 0 and is of course zero for all values of x. Its graph is therefore a horizontal straight line through the origin.
This is the graph of the equation y = 0x+5. This simplifies to y = 5 and so the function has the value 5 for all values of x. It is therefore a straight horizontal line through 5 on the y axis. Play with different values of b and observe the result.
The value of a is 0.5 and b is zero, so this is the graph of the equation y = 0.5x+0 which simplifies to y = 0.5x. This is a simple linear equation and so is a straight line whose slope is 0.5. That is, y increases by 0.5 every time x increases by one. Since the slope is positive, the line slopes up and to the right. Since b is zero, the yintercept is zero and the line passes through the origin (0,0). Play with the a slider and observe the results, including negative values.
The value of a is 0.5 and b is 8, so this is the graph of y = 0.5x+8. The effect of changing b from zero to 8 is that the graph has moved upwards and now passes through 8 on the y axis.