Graphing Square Root and Cube Root Functions

Q: What does the graph of y = √x look like?

It starts at the origin and rises to the right, steeply at first and then more and more slowly. It is half of a parabola lying on its side.

Q: Why is the domain of √x only x ≥ 0?

No real number squares to a negative, so √x has no real value when x is negative. The graph stops at x = 0.

Q: Why can a cube root take negative inputs?

A negative number cubed is negative, so every negative number has a real cube root. ∛(−8) = −2.

Q: Where does y = √(x − h) + k start?

At the point (h, k). The domain is x ≥ h, and when the root has a positive coefficient the range is y ≥ k.

Q: What is the center of a cube root graph?

The point where it changes from bending one way to bending the other. For y = ∛(x − h) + k it is (h, k).

Quick answer

The square root graph is half of a parabola lying on its side: it undoes squaring, and squaring only has an undo for inputs of zero or more. So it starts at a point and climbs ever more slowly. The cube root undoes cubing, which works for every number, so its graph runs in both directions through the origin. Shifts and stretches move the starting point or center.

What you'll learn

Graph y = √x and y = ∛x from a few exact points
Explain the domain and range of each from what it undoes
Graph shifted and stretched radical functions and state their key features

The square root function

f(x) = \sqrt{x}

Plot the inputs whose square roots are whole numbers.

$x$	$0$	$1$	$4$	$9$	$16$
$\sqrt{x}$	$0$	$1$	$2$	$3$	$4$

The inputs spread out while the outputs creep up by one. That is the shape: a steep start, then a curve that keeps rising but more and more slowly.

Domain: $x \ge 0$ . Range: $y \ge 0$ . The graph starts at the origin and has no part to the left of it.

Why the square root graph is half a parabola

$y = \sqrt{x}$ means $y$ is the non-negative number whose square is $x$ :

y = \sqrt{x} \quad\Longleftrightarrow\quad y^2 = x \;\text{ and }\; y \ge 0

So the square root undoes squaring. Swapping the roles of $x$ and $y$ reflects a graph across the line $y = x$ , and that turns the parabola $y = x^2$ into the square root graph.

y = x², x ≥ 0
y = √x
y = x

y = √x is y = x² reflected across y = x

Only the right half of the parabola is used. The left half would pair each positive $x$ with a second, negative $y$ , and a function allows one output per input. That is also why the domain stops at $0$ : no real number squares to a negative, so there is nothing for $\sqrt{x}$ to return when $x < 0$ .

The cube root function

g(x) = \sqrt[3]{x}

The cube root undoes cubing, and cubing has no gaps: every real number is the cube of exactly one real number, negatives included.

$x$	$-8$	$-1$	$0$	$1$	$8$
$\sqrt[3]{x}$	$-2$	$-1$	$0$	$1$	$2$

y = ∛x

The graph of y = ∛x

Domain: all real numbers. Range: all real numbers. The graph runs in both directions and is symmetric through the origin, because $\sqrt[3]{-x} = -\sqrt[3]{x}$ .

Its steepest point is the origin, the center, where the curve switches from bending one way to bending the other.

Shifting and stretching

The usual transformations apply. For the square root:

y = a\sqrt{x - h} + k

Part	Effect
$h$	slides the start point right by $h$
$k$	slides it up by $k$
$a$	stretches vertically; a negative $a$ flips the graph downward

The start point is $(h, k)$ , the domain is $x \ge h$ , and for $a > 0$ the range is $y \ge k$ .

y = 2√(x − 1) − 3

The graph of y = 2√(x − 1) − 3

The cube root works the same way, with a center instead of a start point: $y = a\sqrt[3]{x - h} + k$ has its center at $(h, k)$ , and its domain and range stay all real numbers.

Worked examples

Common mistakes

Practice problems

State the domain and range of $y = \sqrt{x}$ .

Answer

Domain $x \ge 0$ , range $y \ge 0$ .

Full solution

No real number squares to a negative, so inputs must be at least $0$ . The principal square root is never negative, so neither are the outputs.
State the domain of $y = \sqrt{x - 5}$ .

Answer

$x \ge 5$

Full solution

The inside must be zero or more: $x - 5 \ge 0$ , so $x \ge 5$ .
Find the start point of $y = \sqrt{x + 6} + 2$ .

Answer

$(-6, 2)$

Full solution

The inside is zero at $x = -6$ , and there the output is $0 + 2 = 2$ .
Evaluate $\sqrt[3]{-27}$ and $\sqrt[3]{125}$ .

Answer

$-3$ and $5$

Full solution

$(-3)^3 = -27$ and $5^3 = 125$ .
State the domain and range of $y = \sqrt[3]{x}$ .

Answer

All real numbers for both.

Full solution

Every real number has exactly one real cube root, and every real number is the cube root of something — its cube.
Find the center of $y = \sqrt[3]{x - 3} + 2$ .

Answer

$(3, 2)$

Full solution

The inside is zero at $x = 3$ , where the output is $0 + 2 = 2$ .
Find the $x$ -intercept of $y = \sqrt{x} - 3$ .

Answer

$x = 9$

Full solution

$\sqrt{x} - 3 = 0$ gives $\sqrt{x} = 3$ , so $x = 9$ .
Find the $x$ -intercept of $y = 3\sqrt{x + 1} - 6$ .

Answer

$x = 3$

Full solution

$3\sqrt{x + 1} = 6$ gives $\sqrt{x + 1} = 2$ , so $x + 1 = 4$ and $x = 3$ .

Check: $3\sqrt{4} - 6 = 0$ ✓
Explain why $\sqrt{x}$ has no real value for $x < 0$ while $\sqrt[3]{x}$ does.

Answer

Squares are never negative, but cubes of negative numbers are negative.

Full solution

A square root of $x$ is a number whose square is $x$ . Any real number squared is zero or positive, so a negative $x$ has no candidate.

A cube root of $x$ is a number whose cube is $x$ . A negative number cubed is negative, so a negative $x$ always has one: $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$ .
Hana says the range of $y = -\sqrt{x} + 4$ is $y \ge 4$ , because the graph starts at $(0, 4)$ . Find her error.

Hint

Which way does the graph go from its start point?

Answer

The negative sign makes the graph fall from its start. The range is $y \le 4$ .

Full solution

Hana has the start point right. But $-\sqrt{x}$ is never positive, so $-\sqrt{x} + 4$ is never more than $4$ .

As $x$ grows, $\sqrt{x}$ grows and the output falls: at $x = 16$ it is $0$ , at $x = 25$ it is $-1$ .

So the start point is the highest point, and the range is $y \le 4$ .

Frequently asked questions

What does the graph of y = √x look like?