Graphing Radical Functions Calculator

Function setup

Radical type

n (index)

Used when "Nth root" is chosen.

Samples (points)

A (vertical scale)

B (inside scale)

C (horizontal shift)

D (vertical shift)

x-min

x-max

Show data table

Function form:

y = A \cdot root n ( B \cdot (x - C) ) + D

For even n, domain requires B · (x − C) ≥ 0.

Graph

Domain: — Range: —

Computed data

#	x	y

Example data table (y = √x, x = 0…16)

x	y
0	0
1	1
4	2
9	3
16	4

Use this as a quick reference for classic square root growth.

Formula used

We graph functions of the form $y = A \cdot root n ( B \cdot (x - C) ) + D$ where A scales vertically, B scales inside the radical, C shifts horizontally, and D shifts vertically.

Index: n = 2 (square), n = 3 (cube), or any integer n ≥ 2.
Domain: For even n, require B · (x − C) ≥ 0. For odd n, all real x are allowed.
Range: Determined by A, D, and parity of n: for even n and A ≥ 0, range is [D, ∞); if A < 0, range is (−∞, D]. For odd n, range is all real numbers.

How to use this calculator

Select a radical type or choose a custom index.
Set A, B, C, D to control stretch and shifts.
Choose an x range; the tool enforces domain rules.
Click Graph Function to plot and generate data.
Use Download CSV to export the computed dataset.
Use Download PDF to save the chart and summary.
Click Load Example to see a classic transformation.

How to graph a radical function?

A radical of the form $y = A \cdot root n ( B \cdot (x - C) ) + D$ transforms the parent root graph using stretches and shifts.

Identify the index n and its parity (even or odd).
Set the domain: for even n, ensure B · (x − C) ≥ 0; for odd n, any real x is valid.
Locate the key point: the radicand is zero at x = C; the graph passes through (C, D). If B > 0, allowable x are on the side where x ≥ C; if B < 0, use x ≤ C.
Choose sample x-values within the domain and compute y = A · root_n(B(x − C)) + D.
Apply transformations: |A| controls vertical stretch, sign of A reflects vertically; |B| compresses or stretches horizontally, sign of B flips the allowed side.
Plot points and sketch the curve smoothly, noting monotonicity (for many cases, increasing when A·B > 0).
Check intercepts when defined: y‑intercept at x = 0 (if in domain); x‑intercepts from 0 = A · root_n(B(x − C)) + D.

Tip: For y = 2·√(x − 4) − 1, the graph starts at (4, −1) and moves right.

Parameter effects at a glance

Parameter	Effect on graph	Effect on domain	Example
`A` (vertical scale)	Stretches by `\|A\|`; flips over x‑axis if `A<0`.	No change to domain interval.	`A=2` doubles heights; `A=-1` reflects.
`B` (inside scale)	Horizontal compression by `\|B\|`; selects allowed side if even `n`.	Even `n`: if `B>0`, `x≥C`; if `B<0`, `x≤C`.	`B=-1` forces domain to the left of `C`.
`C` (horizontal shift)	Shifts the graph right if larger `C`.	Moves domain boundary to `x=C` for even `n`.	`C=4` starts at `x=4` when `n=2`.
`D` (vertical shift)	Moves the graph up/down.	Even `n`: moves lower/upper bound of range.	`D=-1` lowers the whole curve by one unit.
`n` (index)	Controls curvature; odd roots pass through all y values.	Even `n` restricts domain; odd `n` allows all reals.	`n=3` allows negative radicands without issue.

Domain and range quick guide

Even index, B>0: Domain: x ≥ C, Range: y ≥ D if A≥0, else y ≤ D.
Even index, B<0: Domain: x ≤ C, Range: same bounds using D.
Odd index (any B≠0): Domain: all reals, Range: all reals.

Case A: y = 2·√(x − 1) + 3

Domain: x ≥ 1. Range: y ≥ 3. Key point: (1,3).

Case B: y = -√(5 − x)

Domain: x ≤ 5. Range: y ≤ 0. Key point: (5,0).

Worked examples with domains, ranges, and points

Example 1 (even index): y = 3·√(x + 1) − 2

Domain: x ≥ -1. Range: y ≥ -2. Points: (-1, -2), (8, 7), (15, 10).
Example 2 (even index, B<0): y = -2·√(5 − x) + 4

Domain: x ≤ 5. Range: y ≤ 4. Points: (5, 4), (1, 0), (-4, -2).
Example 3 (odd index): y = 0.5·∛(2(x − 3)) + 1

Domain: all reals. Range: all reals. Points: (3, 1), (5, 1.6299), (-1, 0.3701) (rounded).

Use “Load Example” and adjust parameters to reproduce these cases.