Linear Relations

Lesson

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.

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)

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.

Consider the following.

Which of the following is the graph of a linear relationship?

Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...ALoading Graph...BLoading Graph...CWhat makes that graph linear, but not the other two?

In a linear relationship, the $y$

`y`value is always increasing.AIn a linear relationship, the $y$

`y`value changes at a constant rate.BIn a linear relationship, the $y$

`y`value changes at a faster and faster rate.CIn a linear relationship, the $y$

`y`value is always increasing.AIn a linear relationship, the $y$

`y`value changes at a constant rate.BIn a linear relationship, the $y$

`y`value changes at a faster and faster rate.C

Consider the graph of $y=x-4$`y`=`x`−4.

Loading Graph...

State the coordinates of the $x$

`x`-intercept in the form $\left(a,b\right)$(`a`,`b`).State the coordinates of the $y$

`y`-intercept in the form $\left(a,b\right)$(`a`,`b`).If the graph is translated $6$6 units down, what will be the coordinates of the new $y$

`y`-intercept?State the coordinates in the form $\left(a,b\right)$(

`a`,`b`).

Part of a continuous straight line graph has been graphed. Which of the following statements is true of the *entire* line graph (not just the graphed part).

Loading Graph...

Select all the correct options.

The points on the graph have both positive and negative $y$

`y`values.AThe graph has a negative $x$

`x`-intercept.BThe graph passes through the origin.

CThe points on the graph have only positive $y$

`y`values.DIt has a negative slope.

EIt has a positive slope.

FThe points on the graph have both positive and negative $y$

`y`values.AThe graph has a negative $x$

`x`-intercept.BThe graph passes through the origin.

CThe points on the graph have only positive $y$

`y`values.DIt has a negative slope.

EIt has a positive slope.

F

Identify, through investigation, properties of the slopes of lines and line segments, using graphing technology to facilitate investigations, where appropriate