Coordinates, slopes and reading a chart

Two numbers place a point

A chart has two number lines that cross at zero. The across one is the x-axis and the up-and-down one is the y-axis; where they cross is the origin, (0, 0). Any point is two numbers in a fixed order, (x, y): first how far across, then how far up. The order matters: (2, 5) and (5, 2) are different points. A map works the same way, with longitude across and latitude up, and mixing them up puts a store in the wrong country.

Negative numbers carry on past zero, just as on the number line: negative x is left of the origin and negative y is below it. The axes split the page into four regions, called quadrants:

In Python, a point is often kept as a pair of numbers in brackets, (3, -2), and a set of points as a list of pairs:

points = [(3, 2), (-3, 2), (3, -2), (-3, -2)]
for x, y in points:
    print("across", x, "and up", y)

Slope: rise over run

The slope of a line says how steep it is: how far it goes up for each step across. Between two points, it's the rise (the change in y) divided by the run (the change in x):

slope = (y₂ − y₁) ÷ (x₂ − x₁)

x1, y1 = 1, 3
x2, y2 = 4, 9
slope = (y2 - y1) / (x2 - x1)
print(slope)    # up 6 over 3 steps: 2.0 per step

The sign tells you the direction. A positive slope rises from left to right, a negative slope falls, and a slope of 0 is flat. The size tells you how fast: a slope of 2 rises twice as fast as a slope of 1. Slopes have units, too: if x is days and y is dollars, a slope of 50 means $50 more each day.

The slope between two points far apart is an average rate. Sales that rose $250 over five days rose $50 a day on average, even if most of it came on one busy day. A straight line has the same slope everywhere; a curve has a different slope at each point, and measuring that is exactly what a derivative does in Phase 2.

slope = (400 - 150) / (7 - 2)
print(slope)

week_a, subs_a = 3, 1200
week_b, subs_b = 8, 1800

slope = 0
week_12 = 0

print(slope, week_12)

Lines: y = m × x + b

Any straight line can be written as y = m × x + b. The m is the slope, and b is where the line crosses the y-axis, the value of y when x is 0, called the intercept. It's the formula from the formulas lesson, and the neuron's y = w × x + b with different letters.

Reading a line's formula tells you its shape at once. y = 3x + 5 starts at 5 and climbs 3 for every step across. y = −2x + 4 starts at 4 and falls 2 for every step, so it crosses zero at x = 2. A formula and a chart are two views of the same line, and you can work out one from the other.

To find the formula from two points, get the slope from rise over run, then use either point to find b, because b = y − m × x. Through (1, 3) and (4, 9), the slope is 2, so b = 3 − 2 × 1 = 1, and the line is y = 2x + 1. Checking with the other point: 2 × 4 + 1 = 9. It works.

Reading a chart without being fooled

Before trusting a chart, check four things:

1. The axes and units. What's across, what's up, and in what units? A chart with no units can't be read.
2. Where the y-axis starts. Starting it just below the data makes a 2% change fill the picture. Bars should start at zero.
3. The scale. On a log scale, each step up is ten times the last, so a straight line means steady growth by a percentage, not by an amount.
4. How far the trend is stretched. A line through ten weeks of data says little about week 100, because growth that looks steady over a short stretch rarely stays steady for long.

These questions take seconds, and they catch most misleading charts, including the ones you make yourself.

One more caution: two lines that rise together don't show that one causes the other. Ice-cream sales and sunburns climb together every summer because both follow the weather. A chart can show that things move together; it takes more careful work, like the experiments in Phase 2, to show that one thing causes another.

points = [(3, 2), (-1, 4), (-2, -5), (6, -1), (2, 7)]

top_right = 0
top_left = 0
bottom_left = 0
bottom_right = 0

# loop over the points and count each one in its quadrant

print(top_right, top_left, bottom_left, bottom_right)

Graphs in AI

You'll read these charts constantly. A loss curve plots training steps across and the loss up; its slope shows how fast the model is learning, and a flattening curve means each step helps less than before. A scatter of predictions against true values shows a good model's points hugging the line y = x. A classifier's decision boundary is a line, or a curve, splitting the chart into two regions. And in Phase 1 you'll see words placed as points, where nearby points mean similar meanings.

x = -4
print(2 * x + 3)