A table of values shows a machine's outputs one row at a time. That is fine for five inputs. It is useless for five thousand. A picture can show every input and every output at once, and your eye reads the whole thing in a second. Why do we care? Because from Chapter 1 onward, almost every idea in this course arrives as a picture first: a cloud of dots, a line through them, a curve that bends. If you can put a point on a page and read one back off, you can read all of them.
Think of a graph as a map of the kitchen floor, tiled in squares. To tell a friend where you dropped a coin, you say "two tiles across, seven tiles up." Those two numbers, always in that order, are the point (2, 7). "Across" is measured along the bottom edge of the floor, which math calls the x-axis. "Up" is measured along the left edge, the y-axis. The corner where the two edges meet is the origin, the "zero tiles across, zero tiles up" spot.
Now remember the blender from the last section, f(x) = 2x + 3. Its record book says a 2 went in and a 7 came out. Draw that record as a coin on the floor: 2 across, 7 up. Do it for every row of the record book and you get a trail of coins. That trail is the graph. It is the whole record book in one picture.
Reading the graph is walking the trail backwards. Someone asks, "what comes out when 2 goes in?" Stand on the 2 tile along the bottom edge. Walk straight up until your foot lands on the trail. Now look left to the edge of the floor and read the number: 7. That is it. The only real trap is order. "Two across, seven up" and "seven across, two up" are different tiles, so (2, 7) and (7, 2) are different points.
Take the machine f(x) = 2x + 3 from the last section. Its table says: input 0 gives 3, input 1 gives 5, input 2 gives 7. Now draw each row as a dot. For the row "2 gives 7," start at the corner, walk 2 steps to the right, then 7 steps up, and put a dot there.
A point is one dot on a flat page, named by two numbers written as (x, y). An axis is a number line used as a ruler. The x-axis runs left to right and measures the input. The y-axis runs bottom to top and measures the output. The two rulers cross at the origin, (0, 0). The two numbers naming a point are its coordinates: the first is always the step across, the second is always the step up. So (2, 7) and (7, 2) are different points. A graph of a machine is every point (x, f(x)) drawn at once. For f(x) = 2x + 3 the dots (0, 3), (1, 5), and (2, 7) all sit on one straight path.
Plotting a point means placing it: across first, then up. A negative first number means step left; a negative second number means step down. So (-1, 1) sits one step left of the origin and one step up.
Reading a graph runs the same trip in reverse. To find f(2) from the picture, find 2 on the x-axis, go straight up until you meet the curve, then go straight across to the y-axis and read the number there: 7. This works for any curve, not only straight ones. On the graph of h(x) = x^2, standing at x = 3 and going up meets the curve at height 9, so h(3) = 9.
One limit to know about. A machine with two input slots, like g(x_1, x_2), would need two input axes plus one output axis, three rulers in all. We will rarely draw that. When a picture is needed, this course freezes one input and graphs against the other.
Place two rows of the record book
The grey line is the graph of f(x) = 2x + 3. Drag the two dots onto the points (0, 3) and (2, 7). Across first, then up. Both dots should land exactly on the grey line when you are done.
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Why
Reading a value off a curve that bends
The grey curve is the graph of h(x) = x², which is not a straight path. Put your finger on 3 along the bottom axis, go straight up to the curve, then across to the side axis to read the height. Drag the dot to that spot on the curve, then press Check.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Correct answer
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Why
Why
Worked example
Basic: plot the point (2, 7)
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1
Place the point (2, 7) on a graph and say what it records about f(x) = 2x + 3.
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2across first
The first coordinate is the step across. Start at the origin and move 2 to the right along the x-axis.
across first
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3then up
The second coordinate is the step up. From there move 7 up, parallel to the y-axis, and mark the spot.
then up
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4\boxed{(2, 7)\ \text{sits 2 across and 7 up}}
It records one row of the table: input 2, output 7. Check it against the rule: 2(2) + 3 = 7. The dot belongs on the graph.
Worked example
Typical: five used cars on one graph
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1
A used-car lot lists five cars. Mileage is measured in tens of thousands of miles and price in thousands of dollars: (2, 24), (4, 21), (6, 17), (8, 14), (10, 9). Plot them, then read off the price of the car with 60{,}000 miles.
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2
Choose the axes. Mileage is the input, so it goes across on the x-axis, from 0 to 10. Price is the output, so it goes up the y-axis, from 0 to 25.
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3
Plot each pair across-then-up: 2 across and 24 up, 4 across and 21 up, 6 across and 17 up, 8 across and 14 up, 10 across and 9 up. The five dots step downward from left to right.
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4
To read the price at 60{,}000 miles, find 6 on the x-axis, go up to the dot, then across to the y-axis: \boxed{17,\ \text{so about } \$17{,}000} The picture already tells a story the table hides: more miles, lower price, and the drop is fairly steady. Chapter 1 draws these same five dots again and asks a line to explain them.
Worked example
Common trap: swapping across and up
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1
A student plots the row "input 2, output 7" by walking 7 across and 2 up. Where does the dot land, and why is it wrong?
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2the wrong belief: "the bigger number goes along the bottom"
Walking 7 across and 2 up places the dot at (7, 2).
the wrong belief: "the bigger number goes along the bottom"
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3
Check it against the rule f(x) = 2x + 3. Does input 7 give output 2? No: 2(7) + 3 = 17. The dot at (7, 2) is nowhere near the graph.
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4
The first coordinate is always the input, always the step across, no matter which number is bigger. The correct dot is 2 across, 7 up: \boxed{(2, 7),\ \text{not } (7, 2)} Whenever a plotted dot misses the curve, swapping the two coordinates is the first thing to check.
Where this goes next
This course, Chapter 1 and Chapter 9
Chapter 1, in "Fitting a line by eye," puts the five used cars from the worked example above on a graph and asks you to lay a ruler through the dots. Every judgment in that section is a graph-reading move: which dots sit above the line, which below, and by how much. Chapter 9, "Gradient descent: learning by walking downhill," draws a curve and asks, at a given input, how high the curve is and which way it tilts. Reading a value off a curve, exactly the finger-on-the-axis trip from this section, is the first step of both. Later chapters add a third kind of picture, the "loss curve," which is nothing more than a graph whose input is time and whose output is how wrong the network currently is.
Check your understanding (counts toward your progress)
To plot the point (3, 8), what do you do?
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That plots (8, 3), a different point. The first number is always the step across.
The first number is measured along the bottom axis, across, not up.
The two numbers measure two different directions: across, then up.
Why
Coordinates are always (across, up). The first number is the step along the x-axis, the second the step up the y-axis.
A student plots the row "input 4, output 1" of a table at 1 across and 4 up. Where should the dot have gone?
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That is the swapped position. The input is always the step across.
Only the input is 4; the output is 1.
Only the output is 1; the input is 4.
Why
Inputs go across, outputs go up. Input 4 and output 1 is the point (4, 1).
Which point lies on the y-axis?
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(5, 0) has zero height, so it sits on the x-axis, the bottom ruler.
This point is 5 across and 5 up, away from both axes.
One step across takes the point off the y-axis.
Why
The y-axis is the vertical ruler through the origin, where the step across is 0. Any point (0, y) is on it.
Check your understanding (counts toward your progress)
Five used cars are plotted with mileage across and price up. One dot is at (8, 14). What does it say?
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Mileage was put across, so the first number, 8, is the mileage.
Each dot is one car, not a count of cars.
The two coordinates are two separate facts about one car, never added.
Why
The first coordinate is whatever was put across (mileage), the second is whatever was put up (price). So: mileage 8, price 14.
A dot on a graph does not sit on the curve it was meant to be on. Which is the first thing to check?
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Colour changes nothing about where a point sits.
Axis length changes what is visible, not where a point belongs.
Size is a drawing choice, not a location.
Why
Swapping across and up is by far the most common plotting error, and it moves the dot to a completely different place.
Practice: Points, axes, and reading a curve (5 problems · doesn’t affect your progress)
Which statement about the point (5, 2) is true?
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Why
Coordinates are (across, up), in that order. (5, 2) and (2, 5) are different points.
It is 5 steps across and 2 steps up.
Drag the dot to (3, -4).
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
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A model answer
Why
3 to the right, then 4 down, because the second coordinate is negative.
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Why
On the graph of f(x) = 2x + 3, what height does the line have above x = -3?
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Why
2(-3) + 3 = -3. Above x = -3 the line is actually below the x-axis, 3 steps down.
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Why
The grey curve is h(x) = x^2. Drag the two dots to the points on the curve with inputs x = 2 and x = -2.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
Both inputs square to 4, so the two dots sit at the same height, one on each side of the y-axis.
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Why
Five used cars are plotted as (mileage, price): (2, 24), (4, 21), (6, 17), (8, 14), (10, 9). Reading the graph, how much lower is the price at mileage 10 than at mileage 2?
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Why
At mileage 2 the height is 24; at mileage 10 it is 9. The drop is 24 - 9 = 15.
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