Exercises: Linear functions, zeros, axis intercepts

Look at the graphs of the functions $a (x)$ and $c (x)$ .

Read off the $y$ -axis intercept and the slope of the lines and enter them in the boxes!

Can you work out the function term from this?

What is the $y$ -axis intercept of $a (x)$ ?
For this task you need the following basic knowledge: Line equation in analytic geometry
Determining the $y$ -axis intercept
You determine the $y$ -axis intercept by looking at the intersection of the straight line with the $y$ -axis. In this case:
The $y$ -axis intercept is the $y$ -value of the intersection point $(0 / 4)$ , i.e. $4$ .
Notice where the straight line $G_{a}$ intersects the $y$ -axis.
Determine the intersection.
What is the slope of $a (x)$ ?
For this task you need the following basic knowledge: Slope/Gradient of a line
Determining the slope
The easiest way to determine the slope of a straight line is to use a gradient triangle.
In the case of $G_{a}$ :
You have to go one length unit to the right and one length unit down.
Therefore, you get the slope $m = - 1$
Pick a gradient triangle (see below).
Then determine the slope.
What is the function term of $a (x)$ ?
For this task you need the following basic knowledge: Linear function
Setting up the function term
The function term of a linear function takes the form:
$y = m x + t$
Here $m$ is the slope and $t$ the $y$ -axis intercept.
If you insert the values from the previous sub-tasks, you get:
$y = - 1 \cdot x + 4$
Simplified, that is:
$y = - x + 4$
The function equation of $G_{a}$ is thus:
$a (x) = - x + 4$
.
Recap the basic knowledge about linear functions.
Put your previous values into the function (see below for a more detailed explanation).
What is the y-axis intercept of c(x)?
For this task you need the following basic knowledge: Line equation in analytic geometry
Determining the $y$ -axis intercept
You determine the $y$ -axis intercept by looking at the intersection of the straight line with the $y$ -axis. In this case:
Therefore, the $y$ -axis intercept of $G_{c}$ is $- 3$ .
Notice where the straight line $G_{c}$ intersects the $y$ -axis.
Determine the intersection.
What is the slope of $c (x)$ ?
For this task you need the following basic knowledge: Slope/Gradient of a line
Determining the slope
The easiest way to determine the slope of a straight line is to use a gradient triangle.
In the case of $G_{c}$ :
You have to go one length unit to the right and two length units up.
Therefore, you get the slope: $m = 2$
Pick a gradient triangle (see below).
Then determine the slope.
What is the function term of $c (x)$ ?
For this task you need the following basic knowledge: Linear function
Setting up the function term
The function term of a linear function takes the form:
$y = m x + t$
Here $m$ is the slope and $t$ the $y$ -axis intercept.
If you insert the values from the previous sub-tasks, you get:
$y = 2 \cdot x - 3$
The functional equation of $G_{c}$ is thus:
$c (x) = 2 x - 3$
Recap the basic knowledge about linear functions.
Put your previous values into the function (see below for a more detailed explanation).