"Square root function" is searched about 9,500 times a month in the United States, by Algebra 1 and Algebra 2 students who have just met the graph that is not a line and not a parabola, and need its domain, range and transformations for a test. This page gives the parent graph and its table, the domain and range and why, the half-parabola explanation, the general form with every transformation shown, a four-step sketching method, worked examples, solving square root equations with the extraneous-solution trap, the mistakes, and a printable sheet with practice. Our square numbers and roots page has the arithmetic of roots; this one is about the function.
Download the square root function sheet with practice (PDF) (A4, the parent graph, the table and the transformations on page 1, the sketching method, equations and the practice set with answers on page 2, free).
The parent function: y = √x
| x | 0 | 1 | 4 | 9 | 16 | 25 |
|---|---|---|---|---|---|---|
| y = √x | 0 | 1 | 2 | 3 | 4 | 5 |
Plot those six points and join them with a smooth curve. The graph:
- starts at the origin (0, 0) and does not exist to the left of it;
- rises as x increases (it is an increasing function);
- flattens as it goes: from x = 0 to 1 it climbs 1 unit, from 1 to 4 it climbs 1 unit over 3, from 9 to 16 one unit over 7. The curve keeps rising but ever more slowly;
- has no maximum: √x grows without bound, just slowly (√1,000,000 = 1,000).
Non-square values give decimals: √2 ≈ 1.41, √10 ≈ 3.16, √50 ≈ 7.07. Those are where the curve passes between the plotted points.
Domain and range, and why
Domain: x ≥ 0. In the real numbers you cannot take the square root of a negative number (no real number squares to −4), so the function is only defined for x = 0 and positive x. In interval notation, [0, ∞).
Range: y ≥ 0. The symbol √ means the principal (non-negative) root. √9 is 3, not ±3; the equation x² = 9 has two solutions, but the function √x returns one value, as every function must. So the outputs are never negative. In interval notation, [0, ∞).
Both are written straight onto the graph: the curve starts at the origin and lives in the first quadrant.
Half a sideways parabola: the inverse of y = x²
Square the outputs of y = √x and you get back x: the square root function undoes squaring. That makes it the inverse of y = x², but only of the right half (x ≥ 0), because the full parabola fails the horizontal line test: 3 and −3 both square to 9, and an inverse would have to send 9 to both, which no function can. Reflect the right half of the parabola in the line y = x and the result is y = √x. That is why the graph looks like a parabola lying on its side with the lower half missing. The lower half, y = −√x, is a separate function.
The general form and its transformations
Every square root function can be written y = a√(x − h) + k. The point (h, k) is where the curve starts; a, h and k each move the parent graph:
| Change | Effect | Example | Starting point | Domain | Range |
|---|---|---|---|---|---|
| h > 0 (x − h) | shift right h | y = √(x − 3) | (3, 0) | x ≥ 3 | y ≥ 0 |
| h < 0 (x + h) | shift left | y = √(x + 2) | (−2, 0) | x ≥ −2 | y ≥ 0 |
| k | shift up or down k | y = √x + 4 | (0, 4) | x ≥ 0 | y ≥ 4 |
| a > 1 | vertical stretch | y = 2√x | (0, 0) | x ≥ 0 | y ≥ 0; steeper |
| 0 < a < 1 | vertical compression | y = ½√x | (0, 0) | x ≥ 0 | y ≥ 0; flatter |
| a < 0 | reflect in the x-axis (curve goes down) | y = −√x | (0, 0) | x ≥ 0 | y ≤ 0 |
| negative inside | reflect in the y-axis (curve goes left) | y = √(−x) | (0, 0) | x ≤ 0 | y ≥ 0 |
| all together | y = −2√(x − 1) + 3 | (1, 3) | x ≥ 1 | y ≤ 3 |
Reading domain and range from the form: the domain starts at h (x ≥ h, or x ≤ h if the inside is negated); the range starts at k (y ≥ k if a > 0, y ≤ k if a < 0). No table needed.
A coefficient inside the root, y = √(4x), is a horizontal compression, but since √(4x) = 2√x it is the same curve as a vertical stretch by 2; factor the inside before deciding.
How to sketch any square root function in four steps
Take y = −2√(x − 1) + 3.
- Starting point. h = 1, k = 3: the curve begins at (1, 3).
- Direction. The inside is (x − 1), so the curve goes to the right. a = −2 is negative, so it goes down.
- Three more points. Make the inside 1, 4 and 9: x = 2, 5, 10. The root is 1, 2, 3; times a = −2 gives −2, −4, −6; add k = 3: y = 1, −1, −3. Points (2, 1), (5, −1), (10, −3).
- Draw a smooth curve from (1, 3) through the three points, flattening as it goes. Domain x ≥ 1, range y ≤ 3.
For y = √(x + 4) − 2: start (−4, −2), goes right and up, through (−3, −1), (0, 0), (5, 1). Domain x ≥ −4, range y ≥ −2.
Solving square root equations
- Isolate the root. 2√(x + 3) − 4 = 6 → 2√(x + 3) = 10 → √(x + 3) = 5.
- Square both sides. x + 3 = 25.
- Solve. x = 22.
- Check in the original. 2√25 − 4 = 10 − 4 = 6. Correct.
The extraneous-solution trap. Squaring can introduce answers that do not fit the original. √(x + 3) = x − 3 → x + 3 = x² − 6x + 9 → x² − 7x + 6 = 0 → (x − 6)(x − 1) = 0 → x = 6 or x = 1. Check: x = 6 gives √9 = 3 and 6 − 3 = 3, true. x = 1 gives √4 = 2 and 1 − 3 = −2, false (a square root cannot equal a negative number). Only x = 6. Graphically, y = √(x + 3) and y = x − 3 cross once.
An equation like √(x − 2) = −4 has no solution at all, since the left side is never negative; recognising that before squaring saves a page of work.
Where it is used
The time a dropped object takes to fall h metres is t = √(2h ÷ g); the speed of a wave on a string, the period of a pendulum (T = 2π√(L ÷ g)), the standard deviation in statistics, the distance formula, and the braking distance of a car from its speed all contain square roots. In those models the input must be non-negative for the same reason the domain is x ≥ 0: a negative length or time has no meaning.
The mistakes that cost marks
- Giving the range as "all real numbers": the outputs are y ≥ k (or ≤ k), never both.
- Reading y = √(x − 3) as a shift left 3. Inside the function, the sign is reversed: x − 3 shifts right.
- Forgetting the domain restriction when combining: the domain of y = √(x − 3) is x ≥ 3 even after you add 10 to it.
- Treating √x as ±√x. The function gives one value; the equation x² = a has two.
- Skipping the check after squaring, and keeping an extraneous solution.
- Drawing the curve as a straight line or letting it bend back down. It keeps rising (or falling), ever more slowly.
Where it sits in each curriculum
| Curriculum | When | What is expected |
|---|---|---|
| US | Algebra 1 (graph the parent function, domain and range), Algebra 2 (HSF-IF.C.7b: graph square root functions and show key features; HSF-BF.B.3: transformations; HSA-REI.A.2: solve radical equations and identify extraneous solutions) | the general form, sketching, equations |
| England | GCSE Higher (recognise and sketch y = √x among the standard graphs; transformations of graphs), A-level (inverse functions, domain and range, √x as the inverse of x²) | recognition at GCSE, full treatment at A-level |
| Australia | Year 10 and 10A (graphs of y = √x and transformations), Year 11 Mathematical Methods (functions, inverses) | sketching and domain |
| India (CBSE) | Class 11 (Relations and Functions: domain and range of √x; Sets, Relations and Functions), Class 12 (inverse functions) | domain and range |
Practice (answers in the PDF)
- State the domain and range of y = √(x − 5) + 2 and give its starting point.
- Sketch y = −√(x + 1) and name three points on it.
- Describe the transformations that take y = √x to y = 3√(x − 2) − 4.
- Solve √(2x + 1) = 7.
- Solve √(x + 6) = x. Check for extraneous solutions.
- Why does y = √x have no points in quadrant IV?
Download the sheet and the practice set (PDF)
Where to get help
A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches Algebra 1 and 2, GCSE and A-level maths, the Australian Years 10 to 12 and CBSE Class 11 and 12 every week: functions taught by sketching from the form, with domain and range read off before any table is made, and every radical equation checked in the original. The Algebra 2 page lists the units; the free 30-minute first class is a real lesson on this week's homework. Related pages: square numbers and square roots, solving quadratic equations and exponential growth and decay.
Questions parents ask
1What is the square root function?
2What are the domain and range of the square root function?
3Why is the square root graph half a parabola?
4How do you graph a square root function?
5How do you solve a square root equation?
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