3. Algebraic Relations & Functions

Lesson

We see patterns all around us in the world. From the growth of money in a savings account to the decay of radioactive materials. It can be extremely helpful (and fun) to figure out how to get from one number in a pattern to the next. Knowing how a pattern works can help us make important predictions and plan for the future. A simple pattern (or sequence) is formed when the same number is added or subtracted at each step. Let's take a look at the examples below:

To find the next number that follows in a pattern, it's as simple as figuring out what the pattern is and applying it to the last number. For example, the next number in the decreasing pattern above would be $5-3=2$5−3=2. We could continue this pattern forever if we wanted to!

A table of values can be a nice way to organize a pattern. Below is a drawing of a pattern of flowers.

A table can be generated to count the number of petals visible at a given time, based on how many flowers are present.

Number of flowers |
$1$1 | $2$2 | $3$3 | $4$4 |

Number of petals | $5$5 | $10$10 | $15$15 | $20$20 |

Notice that the number of petals is increasing by $5$5 each time - in particular, the value in the table for Number of petals is always equal to $5$5 times the value for Number of flowers. Therefore, we could generate a rule for this table to say:

$\text{Number of petals}=5\times\text{Number of flowers}$Number of petals=5×Number of flowers

Or to write it more mathematically:

$y=5x$`y`=5`x`

where $y$`y` represents the number of flowers and $x$`x` represents the number of petals.

This rule can now be used to predict future results. For example, to calculate the total number of petals when there are $10$10 flowers present, substitute $x=10$`x`=10 into the rule to find $y=5\times10=50$`y`=5×10=50 petals. So even though there were only $1,2,3$1,2,3 and $4$4 flowers present in the picture above, the rule has determined that there would be $50$50 petals visible when there are $10$10 flowers present.

Let's explore some different patterns in the practice questions below!

Nadia knows that she is younger than her father, Glen. The following table shows her dad's age compared to hers.

Nadia's age | Glen's age |
---|---|

$1$1 | $24$24 |

$5$5 | $28$28 |

$10$10 | $33$33 |

$20$20 | $43$43 |

$30$30 | $53$53 |

- When Nadia was $1$1 year old, Glen was $\editable{}$ years old
- When Nadia is $49$49 years old, Glen will be $\editable{}$ years old.

A catering company uses the following table to work out how many sandwiches are required to feed a certain number of people.

Fill in the blanks:

Number of People |
Sandwiches |
---|---|

$1$1 | $5$5 |

$2$2 | $10$10 |

$3$3 | $15$15 |

$4$4 | $20$20 |

$5$5 | $25$25 |

- For each person, the caterer needs to make $\editable{}$ sandwiches.
- For $6$6 people, the caterer would need to make $\editable{}$ sandwiches.

Consider the pattern shown on this line graph:

Loading Graph...

If the pattern continues on, the next point marked on the line will be $\left(\editable{},\editable{}\right)$(,).

Fill in the table with the values from the graph (the first one is filled in for you):

$x$ `x`-value$y$ `y`-value$0$0 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Choose all statements that correctly describe this pattern:

The rule is $x+3=y$

`x`+3=`y`AAs $x$

`x`increases $y$`y`increases.BThe rule is $y-3=x$

`y`−3=`x`CThe rule is $y+3=x$

`y`+3=`x`DThe rule is $x+3=y$

`x`+3=`y`AAs $x$

`x`increases $y$`y`increases.BThe rule is $y-3=x$

`y`−3=`x`CThe rule is $y+3=x$

`y`+3=`x`D