Generalize y = x² to y = xᵅ — even and odd exponents, domains and symmetry — and a doorway to what comes next.
The bowl y = x² turned out to be just one member of a much bigger clan. Keep the base x but swap the exponent for any constant α, and you have the whole power-function family y = xα: the line x, the bowl x², the S-shaped x³, the square root √x = x½, and the reciprocal 1/x = x−1. Drawn on one set of axes, they reveal patterns you can read at a glance — even versus odd exponents, where each one is allowed to live, which way it climbs. And they set up a quiet hinge into the next stage, where the base will hold still and the exponent will start to move instead.
Everything you learned about y = x² was really about one exponent — the number 2. The graph curls into a bowl because the rule squares the input. But nothing forces that exponent to be 2. Replace it by any constant α and you get a brand-new rule:
A power function is one of the form y = xα, where the base x is the variable and α is a fixed number (the exponent). The bowl y = x² is the case α = 2. Other familiar members come from other choices of α:
α = 1 gives y = x (a line) · α = 2 gives y = x² (a bowl) · α = 3 gives y = x³ (an S) · α = ½ gives y = √x (a root) · α = −1 gives y = 1/x (a reciprocal).
Notice the same machinery met back in Stage 20: x goes in, a single y comes out, y = f(x). What changes from member to member is only the exponent — and, as we are about to see, that one number controls everything about the picture.
The fastest way to understand the family is to meet its five most useful members face to face. Step through them below and watch how each one carries itself — and notice the two that refuse part of the x-axis.
Two members do not live on the whole line. √x has no graph at all left of the y-axis — its domain is x ≥ 0. And 1/x has a hole at x = 0 — its domain is x ≠ 0. Treating either as if it were defined everywhere is the classic slip.
Look again at the bowl x² and the S-curve x³. The bowl is a perfect mirror image of itself across the y-axis; the S-curve is its own image after a half-turn about the origin. That difference is not an accident — it is decided entirely by whether the exponent is even or odd.
An even exponent (x², x⁴, …) makes an even function: f(−x) = f(x). Replacing x by −x gives the same height, so the graph is symmetric about the y-axis (a mirror).
An odd exponent (x, x³, and 1/x = x−1) makes an odd function: f(−x) = −f(x). Replacing x by −x flips the sign of the height, so the graph is symmetric about the origin (a half-turn).
The reason is plain arithmetic: (−x)² = x² (the minus sign cancels), so x² is even; (−x)³ = −x³ (one minus sign survives), so x³ is odd. The same logic stamps every power with the parity of its exponent.
Two more questions finish the portrait of each member: which x does it allow (its domain), and does it rise or fall as you sweep left to right (its monotonicity). Both follow from the exponent. Tracking them in the interval language of Stage 19 and the increasing/decreasing language of Stage 20 keeps the answers tidy.
| Power | Domain | Trend (left → right) | Parity |
|---|---|---|---|
| y = x | all x, (−∞, ∞) | always increasing | odd |
| y = x² | all x, (−∞, ∞) | decreasing then increasing (min at O) | even |
| y = x³ | all x, (−∞, ∞) | always increasing | odd |
| y = √x | x ≥ 0, [0, ∞) | increasing | neither |
| y = 1/x | x ≠ 0 | decreasing on each branch | odd |
Only x² changes direction — it falls to the left of O, bottoms out, and rises to the right. The line x and the S-curve x³ never turn back: they climb everywhere. And 1/x is decreasing on each branch, but you may not say it decreases "everywhere," because it has no value at the gap x = 0.
Here is the hinge that this whole strand has been turning toward. In a power function y = xα, the base x varies while the exponent α stays fixed. In the next family — the exponential y = ax (you will meet it in a later stage) — it is exactly the other way: the base a stays fixed while the exponent x varies. Same two ingredients, opposite roles.
Lay the values side by side and the contrast is vivid:
| x | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| x² (power) | 1 | 4 | 9 | 16 | 25 | 100 |
| 2x (exponential) | 2 | 4 | 8 | 16 | 32 | 1024 |
A power function raises a moving base to a fixed power: y = xα. An exponential function raises a fixed base to a moving power: y = ax. They start out comparable, but for large x the exponential always overtakes any power — its growth compounds. That is the function the next stage will open with. For now, you have met the power clan; you know how to read its shape, its symmetry, its domain, and its climb from one number, the exponent.
The family. A power function is y = xα: a fixed exponent α applied to the variable base x. y = x² is the case α = 2.
The five to know. x (line, all x), x² (bowl, all x), x³ (S, all x), √x = x½ (half-bowl, x ≥ 0), 1/x = x−1 (hyperbola, x ≠ 0). All pass through (1, 1).
Parity. An even exponent ⇒ an even function, f(−x) = f(x), mirror in the y-axis. An odd exponent ⇒ an odd function, f(−x) = −f(x), half-turn about O.
The hinge. In y = xα the base moves and the exponent is fixed; in the exponential y = ax the base is fixed and the exponent moves — and for large x, the exponential always wins.
Written as a power y = xα, what is the exponent α for y = √x?
α = ½. A square root is the same as the ½ power: √x = x½.
Which of y = x, y = x², y = x³ are odd functions?
y = x and y = x³ are odd: their exponents (1 and 3) are odd, so f(−x) = −f(x) and each is symmetric about the origin. y = x² is even (exponent 2), symmetric about the y-axis.
For y = 1/x, which value of x is not allowed, and is the function even or odd?
Not allowed: x = 0 (you cannot divide by zero), so the domain is x ≠ 0. It is odd: 1/x = x−1, the exponent −1 is odd, and 1/(−x) = −(1/x), so it has half-turn symmetry about O.
As x runs from left to right through negative values, does y = x² rise or fall? Explain.
It falls (decreases). The bowl is decreasing on (−∞, 0) — left of its vertex at O — then increasing on (0, ∞). Only at the vertex does it turn around.
Name the domain of y = √x, written as an interval.
x ≥ 0, that is the interval [0, ∞). There is no real square root of a negative number, so the graph stops at the origin and goes no further left.
For large x, which grows faster — x² (a power function) or 2x (an exponential)? Back it up with two values.
2x grows faster. They tie at x = 4 (both 16), but at x = 10, x² = 100 while 2x = 1024 — the exponential has pulled far ahead. For large x an exponential always overtakes any power.
Six questions to lock it in. Tap the answer you think is right.
The big idea. This lesson zooms out from one quadratic to a whole family of power functions y = xα, using the single bowl students already own (x²) as the anchor. The payoff is structural: the exponent α alone dictates shape, domain, symmetry, and monotonicity. Pairing x² with x³ makes even/odd symmetry concrete and visual, and the closing contrast of x² with 2x plants the seed for exponential functions in the next stage — the deliberate "base moves vs exponent moves" hinge.
Watch for these misconceptions. (1) Assuming √x and 1/x are defined for every x — emphasize the domain restrictions x ≥ 0 and x ≠ 0, and that the missing pieces of the graph are not a drawing error. (2) Confusing an even/odd function with an even/odd number — the parity of the function comes from the parity of the exponent, tied to whether the minus sign survives in (−x)α. (3) Confusing a power function xα (variable base, fixed exponent) with an exponential ax (fixed base, variable exponent) — the values table makes the divergence unforgettable.
Common Core. F-IF.C.7b (graphing square-root and other functions), F-BF.B.3 and F-IF.A (even/odd functions and symmetry), N-RN.A (rational exponents preview, √x = x½), F-IF.B.4–5 (domain and behavior — increasing/decreasing, end behavior).