# Slope and Linear Functions

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 \(\Delta y\).”