Non-linear graphs
Lesson

## Linear Vs Nonlinear

### Graphs

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.

### Table of Values

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 $y$yNONLINEAR 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.

#### Examples

##### Question 1

Consider the graph of $y=x^2$y=x2.

1. Which transformation of $y=x^2$y=x2 results in the curve $y=x^2-2$y=x22?

widening the curve

A

reflecting the curve about the $x$x-axis

B

shifting the curve vertically by $2$2 units

C

narrowing the curve

D

shifting the curve horizontally by $2$2 units

E

widening the curve

A

reflecting the curve about the $x$x-axis

B

shifting the curve vertically by $2$2 units

C

narrowing the curve

D

shifting the curve horizontally by $2$2 units

E
2. By moving the graph of $y=x^2$y=x2, sketch a graph of $y=x^2-2$y=x22.

3. What is the equation of the axis of symmetry of $y=x^2-2$y=x22?

##### Question 2

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

1. Complete the table of values for the curve.

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

##### Question 3

Consider the graph of $y=x^2$y=x2.

1. Which transformation of $y=x^2$y=x2 results in the curve $y=\left(x-3\right)^2$y=(x3)2?

shifting the curve horizontally by $3$3 units

A

widening the curve

B

reflecting the curve about the $x$x-axis

C

shifting the curve vertically by $3$3 units

D

narrowing the curve

E

shifting the curve horizontally by $3$3 units

A

widening the curve

B

reflecting the curve about the $x$x-axis

C

shifting the curve vertically by $3$3 units

D

narrowing the curve

E
2. By moving the graph of $y=x^2$y=x2, graph $y=\left(x-3\right)^2$y=(x3)2.

3. What is the axis of symmetry of $y=\left(x-3\right)^2$y=(x3)2?

### Outcomes

#### 9D.LR2.03

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