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Functions with a general form of $y=ax+q$ are called straight line functions. In the equation, $y=ax+q$ , $a$ and $q$ are constants and have different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(x\right)=2x+3$ .
You may have noticed that the value of $a$ affects the slope of the graph. As $a$ increases, the slope of the graph increases. If $a>0$ then the graph increases from left to right (slopes upwards). If $a<0$ then the graph increases from right to left (slopes downwards). For this reason, $a$ is referred to as the slope or gradient of a straight-line function.
You should have also found that the value of $q$ affects where the graph passes through the $y$ -axis. For this reason, $q$ is known as the y-intercept .
These different properties are summarised in [link] .
$a>0$ | $a<0$ | |
$q>0$ | ||
$q<0$ |
For $f\left(x\right)=ax+q$ , the domain is $\{x:x\in \mathbb{R}\}$ because there is no value of $x\in \mathbb{R}$ for which $f\left(x\right)$ is undefined.
The range of $f\left(x\right)=ax+q$ is also $\left\{f\right(x):f(x)\in \mathbb{R}\}$ because there is no value of $f\left(x\right)\in \mathbb{R}$ for which $f\left(x\right)$ is undefined.
For example, the domain of $g\left(x\right)=x-1$ is $\{x:x\in \mathbb{R}\}$ because there is no value of $x\in \mathbb{R}$ for which $g\left(x\right)$ is undefined. The range of $g\left(x\right)$ is $\left\{g\right(x):g(x)\in \mathbb{R}\}$ .
For functions of the form, $y=ax+q$ , the details of calculating the intercepts with the $x$ and $y$ axis are given.
The $y$ -intercept is calculated as follows:
For example, the $y$ -intercept of $g\left(x\right)=x-1$ is given by setting $x=0$ to get:
The $x$ -intercepts are calculated as follows:
For example, the $x$ -intercepts of $g\left(x\right)=x-1$ is given by setting $y=0$ to get:
The graphs of straight line functions do not have any turning points.
The graphs of straight-line functions do not, generally, have any axes of symmetry.
In order to sketch graphs of the form, $f\left(x\right)=ax+q$ , we need to determine three characteristics:
Only two points are needed to plot a straight line graph. The easiest points to use are the $x$ -intercept (where the line cuts the $x$ -axis) and the $y$ -intercept.
For example, sketch the graph of $g\left(x\right)=x-1$ . Mark the intercepts.
Firstly, we determine that $a>0$ . This means that the graph will have an upward slope.
The $y$ -intercept is obtained by setting $x=0$ and was calculated earlier to be ${y}_{int}=-1$ . The $x$ -intercept is obtained by setting $y=0$ and was calculated earlier to be ${x}_{int}=1$ .
Draw the graph of $y=2x+2$
To find the intercept on the y-axis, let $x=0$
For the intercept on the x-axis, let $y=0$
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