Applied exercises: Linear functions

A candle is $18 cm$ long before being lit. When burning, it becomes shorter by $1.5 cm$ every hour.

How many hours does it take for the candle to burn down completely?

For this task you need the following basic knowledge: Lineare Funktionen
Each hour the candle gets shorter by $1.5 cm$ . You get the burning time in hours by calculating how many times the $1.5 cm$ is included in the total length of $18 cm$ : $\frac{18}{1,5} = 12$
Answer: After 12 hours the candle has burned down completely.
Draw the graph of the function $f$ describing the length $y$ (in $cm$ ) of the candle in terms of the burning time $x$ (in hours).

For this task you need the following basic knowledge: Linear function
Graph of $f$
To draw a linear function, you need two points that lie on the graph.
One point is given in the problem setting. At the beginning $x = 0$ the candle is $y = 18$ centimeters long, so $P (0 | 18)$ is a point on the graph.
In part a) you found out that the candle burs down after $x = 12$ hours , i.e. the candle length is then $0 cm$ . So $Q (12 | 0)$ also lies on the graph.
Draw the two points in a coordinate system and connect them.
Find the equation of $f$ .

For this task you need the following basic knowledge: Linear function
Finding the function equation
Examples of the candle length $y$ after a certain burning time $x$ :
At the beginning, $y = 18$ .
After $1$ hour, the length of the candle in $cm$ is:
$y = 18 - 1.5 \cdot 1 = 16.5$
After $2$ hours, the length of the candle in $cm$ is:
$y = 18 - 1.5 \cdot 2 = 15$
After $x$ hours, the length of the candle in $cm$ is:
$y = 18 - 1.5 \cdot x$
So the function equation is $y = - 1.5 \cdot x + 18$
Answer: The equation describing the length of the candle as a function of time is:
$y = - 1.5 \cdot x + 18$
Calculate the length of the candle after 5 or 8 hours. Check your calculated values using the graph.

For this task you need the following basic knowledge: Linear function
Plug $x = 5$ into the function equation.
$y$ $=$ $18 - 1.5 \cdot x$
↓
plug in $x = 5$
$=$ $18 - 1.5 \cdot 5$
$=$ $18 - 7.5$
$=$ $10.5$
Answer: After 5 hours, the candle still has a length of $10.5 cm$ .
Plug x=5 into the function equation.
$y$ $=$ $18 - 1.5 \cdot x$
↓
Plug in $x = 8$
$=$ $18 - 1.5 \cdot 8$
$=$ $18 - 12$
$=$ $6$
Answer: After 8 hours the candle still has a length of $6 cm$ .
After 5 hours, the candle still has a length of $10.5 cm$ . So $R (5 | 10.5)$ must be on the graph.
After 8 hours it still has a length of $6 cm$ . So $S (8 | 6)$ must on the graph.
Since both points $R$ and $S$ indeed lie on the graph of the function $f$ , the candle length was calculated correctly.
Calculate after how many hours, only $3 cm$ of the candle are remaining.

For this task you need the following basic knowledge: Linear function
Plug $y = 3$ into the function equation and solve for $x$ .
$y$ $=$ $18 - 1.5 \cdot x$
↓
plug in $y = 3$
$3$ $=$ $18 - 1.5 \cdot x$ $+ 1.5 \cdot x$
↓
Shift terms with $x$ to the left.
$3 + 1.5 \cdot x$ $=$ $18$ $- 3$
↓
Shift numbers to the right.
$1.5 \cdot x$ $=$ $15$ $: 1.5$
$x$ $=$ $\frac{15}{1.5}$
$=$ $10$
Answer: The candle reaches a length of $3 cm$ after 10 hours.

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$y$	$=$	$18 - 1.5 \cdot x$
	↓	plug in $x = 5$
	$=$	$18 - 1.5 \cdot 5$
	$=$	$18 - 7.5$
	$=$	$10.5$

$y$	$=$	$18 - 1.5 \cdot x$
	↓	Plug in $x = 8$
	$=$	$18 - 1.5 \cdot 8$
	$=$	$18 - 12$
	$=$	$6$

$y$	$=$	$18 - 1.5 \cdot x$
	↓	plug in $y = 3$
$3$	$=$	$18 - 1.5 \cdot x$	$+ 1.5 \cdot x$
	↓	Shift terms with $x$ to the left.
$3 + 1.5 \cdot x$	$=$	$18$	$- 3$
	↓	Shift numbers to the right.
$1.5 \cdot x$	$=$	$15$	$: 1.5$
$x$	$=$	$\frac{15}{1.5}$
	$=$	$10$

Applied exercises: Linear functions

Graph of f

Finding the function equation

Plug x=5 into the function equation.

Plug x=5 into the function equation.

Plug y=3 into the function equation and solve for x.

Graph of $f$

Plug $x = 5$ into the function equation.

Plug $y = 3$ into the function equation and solve for $x$ .