Session 21 - Plotting linear equations

Programs to plot linear equations

Look at the following program that plots equations of the form \(y=mx\).

import numpy as np                      # Import NumPy for numerical calculations
import matplotlib.pyplot as plt         # Import Matplotlib for plotting graphs

x = np.linspace(-6, 6, 400)            # Create 400 x-values from -6 to 6
fig, ax = plt.subplots()               # Create a graph and axis object

for m, label in [(1, r'$y=x$'), (5, r'$y=5x$'), (1/5, r'$y=x/5$')]:
    ax.plot(x, m*x, label=label)      # Plot each straight line y = mx and give it a legend label

ax.scatter([0], [0], color=['tab:orange'], zorder=3)  # Mark the origin (0,0) in orange on top of the lines
ax.axhline(0, color='black', lw=.7)    # Draw the x-axis through y = 0
ax.axvline(0, color='black', lw=.7)    # Draw the y-axis through x = 0
ax.set(xlim=(-6, 6), ylim=(-6, 6), xlabel=r'$x$', ylabel=r'$y$')  # Set the graph limits and axis labels
ax.grid(True, alpha=.3)                # Add a light grid to the graph
ax.legend()                            # Show the legend for the three lines
plt.show()                             # Display the final graph

This program below plots two lines \(y=mx\), marks their rise over run, marks a few points with colored dots, and writes a few names for points and lines etc.

x_values = np.linspace(-1.5, 6, 400)                     # Create 400 x-values from -1.5 to 6 for plotting
fig, ax = plt.subplots(figsize=(7, 5))                   # Create a figure 7 inches wide and 5 inches tall

ax.plot(x_values, 5*x_values, label=r'$y=5x$', color='tab:blue')
# Plot the line y = 5x; label gives the legend text, color makes it blue
ax.plot(x_values, x_values/5, label=r'$y=x/5$', color='tab:orange')
# Plot the line y = x/5; x_values/5 gives the y-values, label sets the legend name, color makes it orange

# Rise and run for y=5x: (0, 0) to (1, 5).
ax.plot([0, 1], [0, 0], '--', color='tab:blue', lw=1.5)
# Draw a dashed horizontal run from x=0 to x=1 at y=0; '--' makes it dashed, color uses blue, lw sets line width
ax.plot([1, 1], [0, 5], '--', color='tab:blue', lw=1.5)
# Draw a dashed vertical rise from (1,0) to (1,5); color is blue, lw sets the line width
ax.annotate('run = 1', (0.5, 0), xytext=(0, -18), textcoords='offset points', ha='center', color='tab:blue')
# Put the text 'run = 1' at (0.5,0); xytext shifts the label by 0 points horizontally and -18 points vertically; offset points means the shift is measured in points; ha centers the text; color makes it blue
ax.annotate('rise = 5', (1, 2.5), xytext=(8, 0), textcoords='offset points', va='center', color='tab:blue')
# Put the text 'rise = 5' at (1,2.5); xytext shifts it by 8 points right and 0 points vertically; textcoords uses points for the shift; va centers the text vertically; color makes it blue

# Rise and run for y=x/5: (0, 0) to (5, 1).
ax.plot([0, 5], [0, 0], '--', color='tab:orange', lw=1.5)
# Draw the dashed horizontal run from x=0 to x=5 at y=0; color uses orange and lw sets the line width
ax.plot([5, 5], [0, 1], '--', color='tab:orange', lw=1.5)
# Draw the dashed vertical rise from (5,0) to (5,1); color is orange, lw sets the line width
ax.annotate('run = 5', (2.5, 0), xytext=(0, -34), textcoords='offset points', ha='center', color='tab:orange')
# Put the text 'run = 5' at (2.5,0); xytext shifts it by 0 points horizontally and -34 points vertically; offset points uses points for the shift; ha centers the label; color makes it orange
ax.annotate('rise = 1', (5, 0.5), xytext=(8, 0), textcoords='offset points', va='center', color='tab:orange')
# Put the text 'rise = 1' at (5,0.5); xytext shifts it by 8 points to the right; textcoords uses point offsets; va centers vertically; color makes it orange

ax.scatter([0, 1, 0, 5], [0, 5, 0, 1], color=['tab:blue', 'tab:blue', 'tab:orange', 'tab:orange'], zorder=3)
# Plot four points at (0,0), (1,5), (0,0), (5,1); the x and y lists give the coordinates; color gives each point a color in order; zorder=3 draws them on top of the lines
ax.annotate(r'$(0,0)$', (0, 0), xytext=(7, 7), textcoords='offset points')
# Label the point (0,0) with text $(0,0)$; xytext shifts the label right by 7 points and up by 7 points; textcoords uses point offsets
ax.annotate(r'$(1,5)$', (1, 5), xytext=(7, 7), textcoords='offset points')
# Label the point (1,5) with text $(1,5)$; xytext moves the label 7 points right and 7 points up
ax.annotate(r'$(5,1)$', (5, 1), xytext=(7, 7), textcoords='offset points')
# Label the point (5,1) with text $(5,1)$; xytext moves the label 7 points right and 7 points up

ax.axhline(0, color='black', lw=.7)  # Draw the horizontal x-axis at y=0; color makes it black and lw sets the line thickness
ax.axvline(0, color='black', lw=.7)  # Draw the vertical y-axis at x=0; color makes it black and lw sets the line thickness
ax.set(xlim=(-1.5, 6), ylim=(-1, 6), xlabel=r'$x$', ylabel=r'$y$')
# Set the x and y limits of the graph; xlabel and ylabel give the names for the axes
ax.grid(True, alpha=.3)  # Add a light grid with transparency 0.3 to make the graph easier to read
ax.legend()  # Show the legend for the two plotted lines
plt.show()  # Display the final graph on screen

Figure 1: Rise and run for calculating the gradient of two lines.

Further work

1. Using the above two programs, hand assemble a program that plots \(y=3x+2\) and \(y=2x+3\). Mark the point of intersection with a dot and label it. Draw rise and run lines and label their lengths.

2. Challenge problem: Can you think of a program that takes the values of m1 and c1 for line 1, m2 and c2 for line 2, plots them and identifies and marks the point of intersection? Write the algebraic logic first, see if you can write the pseudocode.