Analytic Geometry

Lesson

When looking at the cost for services it is important to consider both the upfront fees, also called the fixed cost, as well as the regular payment, also called the variable costs.

Direct variation: Two variables $x$`x` and $y$`y` are said to form a direct variation if they are related through the equation $y=mx$`y`=`m``x` where $m\ne0$`m`≠0. A relationship such as this starts at the origin. In other words, when looking at the cost of services, there is no upfront fee so the dependent variable varies directly with the independent variable.

Partial variation: Two variables $x$`x` and $y$`y` are said to form a direct variation if they are related through the equation $y=mx$`y`=`m``x` where $m\ne0$`m`≠0 and $b\ne0$`b`≠0. A relationship such as this does not start at the origin. In other words, when looking at the cost of services, there is an upfront fee.

Here is a visual to compare the two types of variation.

Type of Variation | Direct | Partial |
---|---|---|

Scenario | A bookkeeper charges $\$65$$65 per hour without any set-up fee. | A bookkeeper charges $\$100$$100 as a set-up fee and then $\$50$$50 per hour. |

Equation | $C=65n$C=65n |
$C=50n+100$C=50n+100 |

Graph |

Which of the following equations represent a direct variation between the pair of variables?

$y=\frac{x}{4},C=0.3n,y=4x+2,C=n+1$`y`=`x`4,`C`=0.3`n`,`y`=4`x`+2,`C`=`n`+1

**Think:** A pair of variables $x$`x` and $y$`y` are said to form a direct variation if they are related through the equation $y=mx$`y`=`m``x` where $m\ne0$`m`≠0. In general, the two variables don't have to be given as $x$`x` and $y$`y`.

**Do:** We can see that in the equation $y=\frac{x}{4}$`y`=`x`4, we have that $m=\frac{1}{4}$`m`=14. Also in the equation $C=0.3n$`C`=0.3`n`, we have that $m=0.3$`m`=0.3. Since these are both non-zero, we claim that $y=\frac{x}{4},C=0.3n$`y`=`x`4,`C`=0.3`n` represent a direct variation between the pair of variables.

**Reflect:** Why are the pair of variables related through the equations $y=4x+2,C=n+1$`y`=4`x`+2,`C`=`n`+1 not a direct variation?

Which of the following represent a partial variation between the variables $x$`x` and $y$`y`? Select all that apply.

Select all that apply.

$y=\frac{3x}{5}$

`y`=3`x`5A$y=x$

`y`=`x`B$y=12+7x$

`y`=12+7`x`C$y=2x+4$

`y`=2`x`+4D$y=\frac{3x}{5}$

`y`=3`x`5A$y=x$

`y`=`x`B$y=12+7x$

`y`=12+7`x`C$y=2x+4$

`y`=2`x`+4D

An outdoor yoga club charges a registration fee of $\$20$$20 plus $\$15$$15 per session that you attend.

Is this payment scheme an example of direct or partial variation?

Direct Variation

APartial Variation

BDirect Variation

APartial Variation

BWrite an equation relating the cost, $C$

`C`, in dollars to the number of sessions attended, $n$`n`.Use your equation to determine the cost for attending $9$9 sessions.

A local climbing club offers two options for the climbing season.

Membership option: A membership fee of $\$54$$54 per year, plus $\$10$$10 per visit.

Casual option: Casual climbing for $\$28$$28 per visit.

Which option is direct variation?

Membership option

ACasual option

BMembership option

ACasual option

BWhat is the initial cost of each option?

Membership option: $\$$$$\editable{}$

Casual option: $\$$$$\editable{}$

Complete the table of values below.

Number of climbing visits in a season 0 1 2 3 4 5 6 Membership option cost (dollars) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Casual option cost (dollars) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ If you are only going to go climbing $2$2 times during a season, which option should you choose?

Membership option

ACasual option

BMembership option

ACasual option

BIf you are going to go climbing $8$8 times during a season, which option should you choose?

Membership option

ACasual option

BMembership option

ACasual option

B

Compare the properties of direct variation and partial variation in applications, and identify the initial value