The graph of $$y = c$$, or

$f(x) = c$

a horizontal line is the graph of a function. We call such a function a constant function. A constant function is one whose output value is the same for every input value. The graph of $$x = a$$ is a vertical line, and $$x = a$$ is not a function.

import matplotlib.pyplot as plt

def constant(x):
return 4

y_values = [constant(x) for x in range(1,10)]

plt.plot(y_values)

The graph of the function given by $$y = mx$$, or

$f(x) = mx$

is a straight line through the origin (0, 0) and the point (1, $$m$$). The constant $$m$$ is called the slope of the line. You could interpret the slope as the common ratio between your variables. Positive slope ($$m > 0$$), lead to lines slanting upwards, flat slopes ($$m = 0$$), lead to horizontal lines, and negative slopes ($$m < 0$$) lead to lines slanting downwards.

def line(m, x):
return m * x

a_values = [line(3, x) for x in range(-10, 10)]
b_values = [line(0, x) for x in range(-10, 10)]
c_values = [line(-1, x) for x in range(-10, 10)]

plt.plot(a_values, label="$f(x) = 3x$")
plt.plot(b_values, label="$f(x) = 0x$")
plt.plot(c_values, label="$f(x) = -1x$")
plt.legend(frameon=False)

In applications where $$m$$ is positive for the equation $$f(x) = mx$$ we say that we have direct variation, and $$m$$ is called the variation constant, or constant of proportionality. Inverse variation is related to the opposite case, where $$f(x) = \frac{m}{x}$$.

A linear function is given by

$f(x) = mx + b$

and has a graph that is the straight line parallel to the graph of $$f(x) = mx$$ and crossing the $$y$$-axis at (0, $$b$$), also called the y-intercept.

def linear(m, b, x):
return (m * x) + b

d_values = [linear(3, 2, x) for x in range(-10, 10)]
e_values = [linear(-3, -1, x) for x in range(-10, 10)]
f_values = [linear(3, 1, x) for x in range(-10, 10)]

plt.plot(d_values, label="$f(x) = 3x + 2$")
plt.plot(e_values, label="$f(x) = -3x -1$")
plt.plot(f_values, label="$f(x) = 3x + 1$")
plt.legend(frameon=False)

A linear function has a couple of forms. The Slope-Intercept Equation form $$f(x) = mx + b$$, and the Point Slope Equation form $$y - y_1 = m(x - x_1)$$. Both of these forms can be used to find the equation of a line, given some prior information (e.g. a couple of points, a slope, an intercept).

The slope of a line containing points ($$x_1$$, $$y_1$$) and ($$x_2$$, $$y_2$$) is

$m = \frac{y_2 - y_1}{x_2 - x_1}$

it’s a change in $$y$$ over a change in $$x$$, which can be read as “for every horizontal change $$\Delta x$$, we have a vertical change $$\Delta y$$.”