Square Root Functions

How domain restrictions and radical graphs behave — and why they start where they do.

A zip line at an adventure park drops from a platform 25 feet high. The engineers need to calculate the speed of a rider at different heights during the descent. The formula they use involves a square root — the speed depends on the square root of the height lost. What struck the engineers right away was that the formula only made sense for heights between zero and 25 feet. You cannot plug in a negative height. That restriction on the inputs is not a quirk of the problem. It is a fundamental feature of square root functions.

The square root function is written as $f(x) = \sqrt$ . The key question is: what values of $x$ can you actually plug in? You cannot take the square root of a negative number and get a real result. So the domain — the set of allowed inputs — is $x \geq 0$ . The graph starts at the origin and moves right and upward, never going left of zero.

f(x) = \sqrt,, \quad x \geq 0

The graph rises quickly at first and then levels off. That is because as $x$ gets large, each additional unit of $x$ adds less and less to $\sqrt$ . From $x = 0$ to $x = 1$ , the output jumps by 1. From $x = 1$ to $x = 4$ , the output only rises from 1 to 2. From $x = 4$ to $x = 9$ , it rises from 2 to 3. The function grows — but slower and slower.

Interactive graph — scroll to zoom, drag to pan

Now consider the general form of a square root function:

f(x) = a\sqrt, + k

Each piece of this formula shifts or stretches the graph in a specific way. The value $h$ shifts the graph left or right — and more importantly, it shifts where the graph begins. The domain becomes $x \geq h$ because you need the expression inside the radical to be non-negative: $x - h \geq 0$ . The value $k$ shifts the graph up or down. The value $a$ stretches it vertically — and if $a$ is negative, the graph reflects across the horizontal axis and opens downward.

The starting point of the graph is always $(h, k)$ . That point is where the expression under the radical equals zero, so KaTeX can only parse string typed expression.

Here is how to graph a transformed square root function step by step.

Graphing f(x) = sqrt(x - 3) + 1

x - 3 \geq 0 \implies x \geq 3

Set the inside of the radical greater than or equal to zero. The graph cannot start before x = 3.

(h, k) = (3, 1)

The starting point is where x = 3 — plug it in and get f(3) = 0 + 1 = 1.

f(4) = \sqrt{4 - 3} + 1 = \sqrt{1} + 1 = 2

Pick a value just to the right of the start to see the curve rising.

f(7) = \sqrt{7 - 3} + 1 = \sqrt{4} + 1 = 3

Another point. The curve rises but slows down.

f(12) = \sqrt{12 - 3} + 1 = \sqrt{9} + 1 = 4 \checkmark

Plot these points and connect them with a smooth curve starting at (3, 1).

Interactive graph — scroll to zoom, drag to pan

Now try one where the graph reflects downward.

Graphing f(x) = -2*sqrt(x + 1)

x + 1 \geq 0 \implies x \geq -1

The expression inside is x + 1, so set it greater than or equal to zero.

(h, k) = (-1, 0)

When x = -1, the radical equals zero and the whole function equals zero. That is the starting point.

f(0) = -2\sqrt{0 + 1} = -2\sqrt{1} = -2

Since a = -2, the graph goes downward from the start.

f(3) = -2\sqrt{3 + 1} = -2\sqrt{4} = -4

Larger x, more negative output. The curve heads down and to the right.

f(8) = -2\sqrt{8 + 1} = -2\sqrt{9} = -6 \checkmark

The graph starts at (-1, 0), curves downward, and keeps falling — never going left of x = -1.

Interactive graph — scroll to zoom, drag to pan

The range — the set of possible output values — depends on $a$ and $k$ . When $a > 0$ , the outputs start at $k$ and go up to infinity: range is $y \geq k$ . When $a < 0$ , the outputs start at $k$ and go down to negative infinity: range is $y \leq k$ .

Practice Questions

f(x) = \sqrt{x + 4}

f(x) = \sqrt{x - 5} + 2

f(x) = -\sqrt{x + 2} - 3

Regents Corner

On Part I of the Algebra I Regents, square root function questions often ask for the domain, the starting point, or which graph matches a given equation. For Part II and Part III, you may need to sketch the graph and label key features. Always state the domain and range using inequality notation — the Regents rubric expects it, and writing just "x ≥ 3" without context or with the wrong variable loses credit.

When the function is f(x) = sqrt(x + 4), students read the "+4" and shift the graph right to (4, 0) instead of left to (-4, 0). The shift direction is opposite to the sign inside the radical. Set x + 4 = 0 and solve: x = -4. That is your starting point, not x = 4.

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Modeling with Quadratics

Transformations of Functions