Identifying key features of Linear functions

The word linear has it's roots in Latin, meaning belonging to a line and a linear function means exactly that.  It is a function that creates the graph of a straight line.  

On a plane a straight line can be drawn in 4 ways.  They can be in any direction and pass through any two points.  

This means that straight lines can be:

 

• An increasing graph means that as $x$x values increase, the $y$y values increase.
• A decreasing graph means that as $x$x values increase, the $y$y values decrease.
• A horizontal graph means that as $x$x values change the $y$y values remain the same
• In a vertical graph the $x$x value is constant. 

Regardless of all different shapes all linear functions have some common characteristics.  

Intercepts

They all have at least one intercept.  Linear functions might have

• an $x$x intercept only (in the case of a vertical line)
• a $y$y intercept only (in the case of horizontal lines) 
• or some have $2$2 intercepts, both an $x$x and a $y$y (in the case of increasing or decreasing functions)

 

Extrema Behaviour

The extrema behaviour of a function is a description of what happens past the viewing zone, what happens with the function outside of the area we can see.  

Take this graph for example, 

We can see in this graph $x$x values between $-5$−5 and $5$5, and $y$y values between $-1$−1 and $6$6.  But we know that the graph goes on and on in the same linear fashion.  

This is the extrema behaviour. All linear functions share the same end behaviour, with a linear function this extrema behaviour is that the line continues forever in the same direction.  No kinks, turns or unexpected movement, just continues in that direction for ever. 

Examples

Question 1

Question 2

Question 3