Most graphs you meet run edge to edge across the page. This one does not. The square root curve begins at a single point, rises to the right, and simply does not exist on the other side — and once you see why, everything else about it follows.
The Parent Curve
The simplest square root function is
Two things stand out immediately.
It starts. At
It flattens. Between
The perfect squares are the points worth plotting:
Why It Is Half a Parabola
Square both sides of
The radical symbol resolves this.
That also makes it the inverse of
Finding the Domain
This is the question exams ask most, and the rule is one line: whatever is under the root must be at or above zero.
Set the inside
| Function | Condition | Domain | Starts at |
|---|---|---|---|
That last row is the one that catches people. Solving
The bracket is square, not round, because zero is allowed under a root —
The Four Transformations
The general form is
and each letter does one job.
moves it horizontally, and the sign flips as usual: shifts right 3, shifts left 4. moves it vertically, with no sign flip: means up 2. stretches it vertically. Larger is steeper. - A negative
reflects it below its starting point, so the curve falls to the right instead of rising.
The starting point lands at
Domain and Range Together
Once you have the starting point
| Case | Domain | Range |
|---|---|---|
| inside is | depends on the sign of |
The range is decided by the sign of
The 1-4-9 Plotting Shortcut
From the starting point, step 1, 4 and 9 units in the direction the curve runs. Those give heights of
For
Worked Example A: Domain and Range
Find the domain and range of
Inside
The start is
Worked Example B: A Left-Opening Curve
Find the domain of
Domain
Worked Example C: A Reflected Curve
Describe
Start at
Domain
Plotting: 1 right and 1 down gives
Worked Example D: A Stretch
Sketch
Start at the origin. The 1-4-9 steps now give heights of
It is the parent curve pulled twice as tall at every point.
Worked Example E: Build the Equation
A square root curve starts at
The start gives
Substituting
Common Mistakes to Avoid
- Giving the domain as all real numbers. The root restricts it; solve the inequality.
- Forgetting to flip the inequality.
gives , not . - Using a round bracket at the start point. Zero is allowed under a root, so it is square.
- Writing
. The radical means the principal root only, . - Getting the horizontal shift backwards.
moves right, not left. - Drawing the curve as a straight line. It flattens as
grows; the gap between plotted points widens. - Restricting a cube root.
accepts negatives and has domain all reals — only even roots restrict.
Where Square Root Functions Lead Next
- Radical equations. Solving
means squaring both sides, which can introduce extraneous solutions that must be checked. - Inverse functions. The root is the inverse of the restricted square, the cleanest example of why domains get restricted at all — see inverse functions.
- Distance and physics. The distance formula is a square root, and so is the period of a pendulum.
- Rational exponents.
connects roots to the exponent rules you already know. - Diminishing returns. The flattening shape models any process where extra input buys progressively less output.
Practice Problems
Work each one before opening the answer.
Problem 1. State the domain of
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Problem 2. State the domain of
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Problem 3. State the domain of
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Problem 4. Where does
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At
Problem 5. What is the range of
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The root is at least 0, so the output is at least
Problem 6. What is the range of
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The reflection makes 1 the maximum. Range
Problem 7. Find
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Problem 8. Give three whole-number points on
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Start
Problem 9. Does
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No. The radical means the principal root, which is never negative.
Problem 10. State the domain of
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All real numbers,
Problem 11. Describe how
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Shifted 3 left and 4 down, so it starts at
Problem 12. A square root curve starts at
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Quick Reference
| Task | Method |
|---|---|
| General form | |
| Starting point | |
| Domain | set the inside |
| Bracket at the start | square — zero is allowed |
| Range, | |
| Range, | |
| Horizontal shift | |
| Reflection | negative |
| Plotting shortcut | step 1, 4, 9 from the start |
| Negative coefficient on | curve opens leftward |
A square root graph is half a sideways parabola, and it is the inverse of the squaring function once that has been restricted to be one-to-one. Its main exam question is the domain, and the algebra underneath is covered in radicals. To drill the domain step, try the domain and range practice problems. More Algebra lessons are available.