Linear functions

Basic functions: Linear functions

Linear functions

By convention $f$ is called a linear function (another term for first degree function) in $x$ if the function definition can be written as

$f(x)=a\,x+b,$ for certain numbers $a\neq 0$ and $b$ .

The graph of $f(x)=a\,x+b$ originates from the graph of the function $x\mapsto x$ by vertical multiplication with respect to the $x$ -axis followed by a vertical translation.

The graph of $f(x)=a\,x+b$ is a straight line with gradient or slope $a$ , and vertical intercept $b$ .

GeoGebra

For the first degree function $f(x)=a\,x+b$ the following is true:

With an increase in $h$ length units in the horizontal direction along the graph corresponds an increase in $a\cdot h$ length units in the vertical direction.
The graph of the function $f$ intersects the vertical axis at the point $(0,b)$ because at $x=0$ the function value equals $f(0)=b.$

GeoGebra

Horizontal intercept? The intercept of a graph of a function is the vertical coordinate of the intersection of the graph with the vertical axis.

The reason that this notion is limited to the vertical axis is the the intersections of a graph of a function with the horizontal axis, i.e., zeros of a functions, are not uniquely determined; there can be no zero or more than one zero

Below is the graph of the linear function

$C(t)=-\frac{2}{3}t+2$ in which we deliberately use variables other than the usual $f$ , $x$ , and $y$ . You can think of a concentration that decreases linearly in the course of time (for example, the blood concentration of alcohol in the human body after alcohol consumption).

graph of a linear function

The slope can be found in the graph. The slope is in fact the ratio of the increment ${\vartriangle}C$ of $C=C(t)$ and the increment ${\vartriangle}t$ of $t$ . For example, between $t=0$ and $t=3$ the increment of $t$ is equal to ${\vartriangle}t=3-0=3$ , and the increment of $C=C(t)$ is equal to ${\vartriangle}C=0-2=-2$ . The quotient is $\displaystyle \frac{{\vartriangle}C}{{\vartriangle}t }=\frac{-2}{3}=-\frac{2}{3}$ .

TThe intercept with the vertical axis can be read in this case directly in the graph because the function value at $t=0$ is visible. If this is not the case, then you must read another point, say $(C_1,t_1)$ , in the graph and calculate the intercept as $C_1-a\cdot t_1$ , under the assumption that you have already determined the slope $a$ before.

Application in image processing An application of linear functions in image processing is a linear change in brightness and contrast by applying the function

$x\mapsto c\cdot x+b$ on the light intensity of each pixel with positive contrast parameter $c$ (strengthening if as $c>1$ , weakening as $0<c<1$ ) and with brightness parameter $b$ (brighter as $b>0$ , darker as $b<0$ ).

The receptive field of visual neuron is the area in the visual field where the firing pattern changes when it is stimulated by certain visual stimuli. Assume that the response of a nerve cell to one visual image $I_1$ is a firing pattern with frequency $r_1=20\;\text{Hz}$ and that the response to another visual image $I_2$ a firing pattern with frequency $r_2=40\;\text{Hz}$ . In a linear stimulus-response model, the response of this cell to the average stimuls $\tfrac{1}{2}I_1+\tfrac{1}{2}I_2$ will be a firing pattern with frequency $\tfrac{1}{2}r_1+\tfrac{1}{2}r_2=30\;\text{Hz}$ . If one stimulus, say $I_2$ is kept constant and the other stimulus $I_1$ varies, we get in this model a linear function of the stimulus to the firing frequency .

Please note that this is a mathematical model only, because this model could result in an unrealistically high firing frequency in response to a strong stimulus.

Mathcentre video

Linear Functions (18:56)