Applied exercises: Linear functions

Are the following relations are linear, proportional or non-linear?

number of bought cucumbers $\mapsto$ total price of those cucumbers
(assumed that there is no discount)
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
The relation is linear and even proportional.
The price of cucumbers is calculated per piece. Therefore, twice as many cucumbers cost twice as much and five times as many cucumbers cost five times as much.
Example
The graph of the price depending on the cucumber nurmber could look as follows:
age $\mapsto$ body height
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
This relation is not linear, i.e. in particular not proportional.
Until the age of about 18, people are still growing. After that, growth stops. Therefore, the relation cannot be linear.
Example
This is what the body height of a person over a lifespan may look like:
(The graph is simplified)
amount of rice $\mapsto$ weight of a bowl with that rice in it
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
This relation is linear, but not proportional.
The weight of the filled bowl increases linearly with the amount of rice. However, since the bowl has its own weight, the relation is not proportional. The filled bowl does not weigh twice as much with double the amount of rice.
Example
For a bowl of $200 g$ , the weight function may look as follows:
value in € $\mapsto$ value in $
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
The relation is linear and even proportional.
The exchange rate between € and $ indicates how much $ you get for 1 €. The more monetary value you have in €, the higher its monetary value in $. Twice as much value in € means twice as much value in $. Ten times as much value in € means ten times as much value in $, etc. The relation is therefore linear and even proportional.
Example
This is what a money conversion function may look like:
study time for an exam $\mapsto$ score in the exam
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
This relation is not linear, so in particular also not proportional.
The more you study, the more likely it is that you will get a better grade and more points in the exam. However, after a certain period of study, you may have learned so well that you get (almost) full score in the exam. Further learning will then not lead to (much) further improvement.
Example
The graph of an associated function could look like this:
side length of a square $\mapsto$ area of the square
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
This relation is not linear, so in particular also not proportional.
The area of a square with side length a is calculated via $A_{□} = a^{2}$ . Doubling the side length leads to a quadrupling of the area. A tripling of the side length leads to a ninefold increase in the area.
You can also use further examples, as in the table of values below.
The relation is thus not linear and not proportional.
Example
For example, take a square with a side length of 1 LU (length unit), 2LU, etc.. Then calculate the area of the square in AU (area units).
a
1 LU
2 LU
3 LU
4 LU
5 LU
$A_{□}$
1 AU
4 AU
9 AU
16 AU
25 AU
The graph of an associated function could look like this:
number of chocolate bars $\mapsto$ number of calories gained by the chocolate bars
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
The relation is linear and even proportional.
Each chocolate bar has a certain number of calories. If you eat twice as many chocolate bars, you eat twice as many calories as if you eat one chocolate bar. Three bars triple the calories, etc.
Example
The graph of an associated function could look like this:
number of drinks during a club visit $\mapsto$ total costs of the club visit
(All drinks cost the same.)
- linear
- proportional
- non-linear
For this task you need the following basic knowledge: Linear function
This relation is linear, but not proportional.
You have to pay an entrance fee to get into a club. In addition, you pay a certain amount for each drink. Therefore, the costs have to be calculated as follows:
$cost = entrance fee + number of drinks \cdot price per drink$
Due to the entrance fee, the relation is not proportional.
Example
The graph of an associated function could look like this: