Recognising cubic, reciprocal and exponential graphs
Describe the shapes and key features of y = x³, y = 1/x and y = kˣ, calculate their values, and match fixed-multiplier situations to exponential models.
What a learner can do afterwards
- Describe cubic and reciprocal graph features, calculate points, and explain the reciprocal's excluded input and asymptotes
- Identify a fixed multiplicative factor as exponential change, and calculate or compare values over stated steps
- Apply a stated shift or reflection to one of these shapes
1 · Read
The shifts, stretches and reflections you have learned also work on cubic, reciprocal and exponential graphs. Plot (x, y) from the origin: move horizontally to x, then vertically to y, using signed coordinates. Choose a labelled scale for each axis. Study each shape's points, symmetry and behaviour near the axes to sketch a smooth curve and match it to a rule or situation. Adding a constant shifts vertically; replacing x with x - a shifts horizontally.
For y = x³, use three equal factors: x times x times x, with two multiplications. The curve increases through the origin and has a flat spot there. It has half-turn symmetry: (2, 8) pairs with (-2, -8). It is steep far from zero on either side and flatter near zero. On the negative side, it becomes less steep as x approaches zero; on the positive side, it becomes steeper as x grows. Recover x by finding the number whose cube is y. Check by cubing.
For y = 1/x, x cannot be zero and y never equals zero. Its branches lie in the first and third quadrants. Near zero, y increases without bound from the right and decreases without bound from the left. Far from zero in either direction, y approaches zero. The axes are asymptotes. Doubling a nonzero input halves its output. For 1/(x - a), the vertical asymptote is x = a, where the denominator is zero. Recover x from nonzero y by solving xy = 1: x = 1/y.
The curve approaches the y-axis without touching it.
For y = 2^x, each increase of 1 in x doubles y. The curve is positive, passes through (0, 1), approaches zero to the left and rises faster to the right. A negative power reverses doubling: 2^-2 = 1/4. For y = k^x, k > 1 gives growth and 0 < k < 1 gives decay; both pass through (0, 1). A fixed positive starting amount multiplied by the same factor each step follows an exponential model.
To sketch y = x³, cube several inputs: -2, -1, 0, 1 and 2 give -8, -1, 0, 1 and 8. Plot the points and draw a smooth S-curve through the origin, flat there and steeper farther away. Check half-turn symmetry: (2, 8) pairs with (-2, -8). The points (3, 27) and (-3, -27) show how steep the tails become.
To sketch y = 1/x, use points (1, 1), (2, 1/2), (4, 1/4) and (8, 1/8). Larger positive inputs give outputs closer to zero. Negative inputs give the other branch: (-1, -1) and (-2, -1/2). Near zero from the right, inputs 1, 1/2, 1/4 and 1/8 give outputs 1, 2, 4 and 8. Draw two separate smooth branches approaching the axes, with no point at x = 0.
Find a y-intercept by setting x = 0. For y = 4^x + 3, 4^0 = 1, so y = 4 and the intercept is (0, 4). A positive base raised to zero is 1. An added constant changes the output, including the intercept; a number inside the exponent changes the input instead.
A fixed ratio identifies repeated multiplication: 2, 8, 32, 128 uses factor 4 each step. Adding 5% of a current amount B gives B + 0.05B = 1.05B, the same multiplier each year. To compare increases over a stated interval, evaluate both endpoints and subtract. For 3^x from x = 1 to 2, the increase is 9 - 3 = 6. Growing differences alone do not prove a model is exponential.
Use the transformation rules on these functions too. Adding k to f(x) raises every output by k. Replacing x with x - h shifts corresponding points right h. A minus sign outside reflects in the x-axis: the new output is the negative of the old output. For y = -x³ at x = 10, first calculate 10³ = 1000, then negate it to get -1000. For y = 1/x + 3 at x = 4, find 1/4 = 0.25, then add 3 to get 3.25.
The cubic is an increasing S-curve through the origin. The reciprocal has two branches approaching the axes. For k > 1, k^x is positive, approaches zero to the left and grows to the right.
The practice questions and reading were written for this dataset. The referenced video is by Khan Academy.
More to read
Growth, decay and reflection
For y = b^x with b > 0 and b not equal to 1, increasing x by 1 multiplies the output by b. A base above 1 gives growth; a base between 0 and 1 gives positive exponential decay. Both rules pass through (0, 1), since a positive base raised to zero is 1.
For y = 3^x, reflecting in the x-axis gives y = -3^x. As x increases, these outputs become more negative and their magnitudes grow. This is a reflection of growth, not positive exponential decay.
A vertical translation by k moves the horizontal asymptote from y = 0 to y = k. You can find a translated y-intercept by substituting x = 0. Keep changes to the exponent separate from changes added outside the power.
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3 · Play
Plot the points yourself, then let the curve appear through them.
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
36 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.