Recognising Cubic, Reciprocal and Exponential Graphs
Recognise and sketch the shapes of y = x³, y = 1/x and y = kˣ, and match each shape to the kind of situation it describes.
What a learner can do afterwards
- Sketch y = x³ and y = 1/x, including the axes the reciprocal graph never touches
- Pick the exponential curve rather than a straight line for a doubling story
- Apply a stated shift or reflection to one of these shapes
1 · Read
The last stop moved familiar graphs with shifts, stretches and reflections. Those moves carry over: a number added to the rule slides a curve up, and a number subtracted from x slides it sideways. This stop meets three new shapes you will keep seeing: the cubic, the reciprocal and the exponential. The skill is to sketch each one from memory, say what happens at the tricky spots, and match each shape to the kind of situation it describes. You do not need to draw them perfectly. You need to know what they look like, where they bend, and where they refuse to go.
Start with the cubic, y = x cubed. It passes through the origin and bends there, with a single flat spot where the curve levels off for an instant before climbing again. The shape is odd: turn it a half turn around the origin and it lands on itself, so the point (2, 8) has a partner at (-2, -8). Near zero the curve is flat and lazy, but out in the tails it climbs steeply, because cubing a big number makes it much bigger. The larger the x, the faster the y grows, on both sides.
The reciprocal, y = 1/x, has two separate branches that never meet. In the first quadrant the curve sits high and close to the y-axis, then trails off toward the x-axis as x grows. The third quadrant holds the mirror branch, low and negative. The axes are the lines it never touches. As x gets close to zero, 1/x shoots off toward infinity, and as x gets huge, 1/x shrinks toward zero. So the x-axis and y-axis are the asymptotes, the borders the curve approaches but never crosses.
Getting closer to the border, never arriving.
The exponential, y = 2 to the power of x, is the curve of steady doubling. It stays above the x-axis no matter how far left you go, because a positive base to any power stays positive. It passes through (0, 1), since anything to the power zero is 1. To the left the curve hugs the x-axis and looks almost flat, and to the right it climbs ever faster. That is why a doubling story, a spreading rumour, interest that compounds, draws an exponential rather than a straight line.
Sketch y = x cubed. Pick x values and cube them: at x = -2, y = -8; at x = -1, y = -1; at x = 0, y = 0; at x = 1, y = 1; at x = 2, y = 8. Plot them and the S-curve appears, bending through the origin. Check the odd symmetry: the point (2, 8) should have a partner at (-2, -8), and it does. Now the tails: at x = 3, y = 27, and at x = -3, y = -27. The curve is pulling away fast, the steep climb the shape promised.
Now sketch y = 1/x with a small table. At x = 1, y = 1; at x = 2, y = 1/2; at x = 4, y = 1/4; at x = 8, y = 1/8. As x grows, y gets smaller, sliding toward the x-axis without ever reaching it. On the negative side, x = -1 gives y = -1, and x = -2 gives y = -1/2. The tricky spot is near zero. As x gets close to 0 from the right, 1/x climbs: 1, 2, 4, 8, and beyond, shooting up toward the y-axis without touching it. That is the asymptote in action.
Sketch the cubic as an odd S-curve through the origin, the reciprocal as two branches that never touch the axes, and the exponential as an always-positive curve that hugs the axis on the left and doubles on the right.
2 · Watch
3 · Play
Plot the points yourself, then let the curve appear through them.
Take it off screen
Where it sits
Learn first
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.