Analytic Geometry

Lesson

A linear relationship is a relationship that has *constant rate of change*. The slope is a constant value and the $y$`y` values change by the same amount for constant changes in $x$`x` values.

Linear relationships, when graphed, are **STRAIGHT LINES**!

This makes anything that is not a straight line nonlinear.

These graphs are all linear.

These graphs are all nonlinear.

As we saw in the previous lesson on tables of values, identifying if a function is linear from a table of values requires us to check for a constant rate of change in the $y$`y`-values.

Here are some examples:

Constant change in $x$`x` and in $y$`y` LINEAR RELATIONSHIP

Constant change in $x$`x`, not a constant change in $y$`y`, NONLINEAR RELATIONSHIP

Constant change in $x$`x` and in $y$`y` LINEAR RELATIONSHIP

Non constant change in $x$`x`, non constant change in $y$`y`. Would need to check if Linear by checking the slope formula. This in fact is Linear - can you find the rule?

Non constant change in $x$`x`, non constant change in $y$`y`, would need to check using the slope formula. This is NONLINEAR.

Consider the graph of $y=x^2$`y`=`x`2.

Loading Graph...

Which transformation of $y=x^2$

`y`=`x`2 results in the curve $y=x^2-2$`y`=`x`2−2?widening the curve

Areflecting the curve about the $x$

`x`-axisBshifting the curve vertically by $2$2 units

Cnarrowing the curve

Dshifting the curve horizontally by $2$2 units

Ewidening the curve

Areflecting the curve about the $x$

`x`-axisBshifting the curve vertically by $2$2 units

Cnarrowing the curve

Dshifting the curve horizontally by $2$2 units

EBy moving the graph of $y=x^2$

`y`=`x`2, sketch a graph of $y=x^2-2$`y`=`x`2−2.Loading Graph...What is the equation of the axis of symmetry of $y=x^2-2$

`y`=`x`2−2?

Consider the curve whose equation is $y=\left(x+4\right)\left(x+2\right)$`y`=(`x`+4)(`x`+2).

Complete the table of values for the curve.

$x$ `x`$-4$−4 $-2$−2 $-1$−1 $y$ `y`$\editable{}$ $\editable{}$ $\editable{}$ Use the points in the table to sketch the curve.

Loading Graph...

Consider the graph of $y=x^2$`y`=`x`2.

Loading Graph...

Which transformation of $y=x^2$

`y`=`x`2 results in the curve $y=\left(x-3\right)^2$`y`=(`x`−3)2?shifting the curve horizontally by $3$3 units

Awidening the curve

Breflecting the curve about the $x$

`x`-axisCshifting the curve vertically by $3$3 units

Dnarrowing the curve

Eshifting the curve horizontally by $3$3 units

Awidening the curve

Breflecting the curve about the $x$

`x`-axisCshifting the curve vertically by $3$3 units

Dnarrowing the curve

EBy moving the graph of $y=x^2$

`y`=`x`2, graph $y=\left(x-3\right)^2$`y`=(`x`−3)2.Loading Graph...What is the axis of symmetry of $y=\left(x-3\right)^2$

`y`=(`x`−3)2?

Identify, through investigation, some properties of linear relations, and apply these properties to determine whether a relation is linear or non-linear