A new member of the function family: when the highest power is a square, y = ax² + bx + c (a ≠ 0).
You have already met functions that turn an input into an output by a steady rule — double it, add to it, share it out. But some things grow faster the bigger they get: widen a square photo and its area balloons far quicker than its side. That “side squared” relationship is the first whisper of a new kind of function — the quadratic. This lesson welcomes it into the family, writes its definition y = ax² + bx + c, teaches you to spot a, b, and c, and plots the simplest one so you can see its shape.
A function is one rule that turns each input x into a single output y, written y = f(x). You have already kept company with a few of them — and each one drew its own shape on the coordinate plane (and you learned to read those shapes back in functions and graphs):
Each of these is one rule, one graph. But notice what they all have in common: the input x only ever appears to the first power. Now meet a rule that does something new — it squares the input. As soon as an x² shows up, the line bends into a curve, and we have a brand-new member of the family.
A function is one rule, y = f(x), turning each input into a single output. Lines and gentle curves you have met all keep x to the first power. Squaring the input is the new move — and it bends the graph into a parabola.
Here is the squaring rule in disguise. A square photo of side x has area S = x². Grow the side one step at a time and watch the area:
| side x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| area S = x² | 1 | 4 | 9 | 16 |
| jump in area | — | +3 | +5 | +7 |
The side climbs in equal steps — 1, 2, 3, 4 — but the area leaps unevenly: +3, then +5, then +7. Each extra unit of side adds more area than the last. That accelerating, uneven growth is the signature of squaring, and it is exactly why the graph of S = x² curls upward instead of staying straight.
When the output depends on the square of the input, growth speeds up the larger x gets, and the graph curves rather than running straight. That is the heart of a quadratic.
Tidy any such rule into standard shape and it always reads the same way. A quadratic function is any rule that can be written as
y = ax² + bx + c (a, b, c constants, a ≠ 0)
The ax² term is what makes it quadratic — it is the highest power, and it is the part that bends the graph. The three constants have names you will use again and again:
a ≠ 0 is non-negotiable. If a = 0, the x² term vanishes and you are left with y = bx + c — an ordinary straight line, not a parabola. The square must survive for the function to be quadratic.
To name a, b, and c, first tidy the expression into the standard shape ax² + bx + c — expand any brackets, collect like terms, write the powers in order — then simply read off the numbers. Missing terms count as 0. Here are four worked tidy-ups:
| expression | standard form | a | b | c | quadratic? |
|---|---|---|---|---|---|
| y = 3x² − 5x + 2 | 3x² − 5x + 2 | 3 | −5 | 2 | yes |
| y = x² | x² + 0x + 0 | 1 | 0 | 0 | yes |
| y = 2 − x² | −x² + 0x + 2 | −1 | 0 | 2 | yes |
| y = (x + 1)(x − 2) | x² − x − 2 | 1 | −1 | −2 | yes |
| y = 3x + 2 | 0x² + 3x + 2 | 0 | 3 | 2 | no — linear |
| y = 1/x + 1 | not a polynomial | — | — | — | no |
The two red rows are non-examples: y = 3x + 2 has a = 0, so it is linear; y = 1/x + 1 has x in the denominator, so it is not even a polynomial — there is no x² to call a at all.
Read off a, b, c for y = (x + 1)(x − 2). Expand: (x + 1)(x − 2) = x² − 2x + x − 2 = x² − x − 2. So a = 1, b = −1, c = −2, and since a ≠ 0 it is quadratic.
Set a = 1, b = 0, c = 0 and you get the plainest quadratic of all, y = x² — square the input, nothing else. Make a small table of values for x from −3 to 3:
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y = x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
Plot those seven points and join them with a smooth sweep, and you get a graceful bowl — the parabola. Two things jump out:
y = x² is the parent parabola. Its lowest point is the origin (0, 0), and it folds perfectly about the y-axis. Every other quadratic graph is just this same bowl reshaped, flipped, and slid around the plane — which is exactly the story of the next lesson.
Is y = 4x² − 1 a quadratic function? If so, name a, b, and c.
Yes — there is an x² term and its coefficient is not 0. In standard form it is 4x² + 0x − 1, so a = 4, b = 0, c = −1.
Is y = 5x − 2 a quadratic function? Explain.
No. There is no x² term, so a = 0 — this is a linear function (a straight line), not a quadratic.
Expand y = (x − 3)(x + 1) into standard form and give a, b, c.
(x − 3)(x + 1) = x² + x − 3x − 3 = x² − 2x − 3, so a = 1, b = −2, c = −3.
For y = x², find y when x = −4, and when x = ½.
y = (−4)² = 16; y = (½)² = ¼. (Note (−4)² is positive — squaring removes the minus sign.)
A square has side s. Write its area as a function of s, and find the area when s = 7.
Area A = s² (a quadratic with a = 1, b = 0, c = 0). When s = 7, A = 7² = 49.
The points (2, 4) and (−2, 4) both lie on y = x². What does that tell you about the graph?
Two points at the same height y = 4, equally far from the y-axis (at x = 2 and x = −2), show the graph is symmetric about the y-axis — because (−x)² = x², every input x and its opposite −x give the same output.
Six questions to lock it in. Tap the answer you think is right.
The big idea. This lesson introduces the quadratic function y = ax² + bx + c (a ≠ 0) as the next member of the function family, after the linear and proportional functions of Stages 20–21. The two anchors are concrete: area grows like the square of the side (uneven, accelerating growth → a curved graph), and the parent parabola y = x² (symmetric about the y-axis, lowest point at the origin). Students should leave able to recognize a quadratic, tidy it into standard form, and name a, b, c. Later families — inverse proportion (Stage 22) and exponential and logarithmic functions (Stage 24) — will extend this same function-machine idea.
Misconceptions to watch. (1) Forgetting that a ≠ 0 — students sometimes call any equation with letters “quadratic.” Stress that the x² term must survive. (2) Confusing b and c, or missing that an absent term means a 0 coefficient (in y = 4x² − 1, b = 0). (3) Believing (−x)² = −x² — emphasize that squaring kills the sign, which is exactly what makes the bowl symmetric and non-negative. (4) Thinking any curved-looking graph is a parabola; the hyperbola y = k/x curves too but is a different family entirely.
Common Core. F-IF.A.1–2 (function notation and evaluation), A-SSE.A.1 (interpreting the parts a, b, c of an expression), F-IF.C.7a (graphing quadratics, intercepts), and F-BF.A.1 (building a function from a situation — here, area as a function of side).