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Linear Equations
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Key Stage 3

Recognise linear and non-linear relationships

Lesson
Practice
Lesson

Linear Vs Nonlinear

Graphs

A linear relationship is a relationship that has constant rate of change.  The gradient 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 $y$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 gradient 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 gradient formula. This is NONLINEAR.

Examples

Question 1

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

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

    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=x2−2.

    Loading Graph...

  3. What is the equation of the axis of symmetry of $y=x^2-2$y=x2−2?

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.

    Loading Graph...

Question 3

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

Loading Graph...

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

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

    Loading Graph...

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

 

 

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