Square Root Functions

1 General Form & Parameters

📐

y = a√(b(x − h)) + k
This is the standard form of all square root functions. Every square root question can be solved by identifying these four parameters.

What Each Parameter Controls

Parameter	Name	Effect on Graph
a	Vertical scale & direction	a > 0 → graph goes UP from vertex \| a < 0 → goes DOWN
b	Horizontal direction	b > 0 → graph goes RIGHT \| b < 0 → goes LEFT
h	Horizontal shift	Vertex x-coordinate (from −h in the formula, so reverse sign)
k	Vertical shift	Vertex y-coordinate → vertex = (h, k)

💡

Vertex = (h, k) — always read from the standard form after factoring.

2 The Four Basic Graphs

The signs of a and b determine which "corner" the graph opens toward:

a sign	b sign	Direction from vertex	Example
+	+	Up and Right (→↑)	y = √x
−	+	Down and Right (→↓)	y = −√x
+	−	Up and Left (←↑)	y = √(−x)
−	−	Down and Left (←↓)	y = −√(−x)

3 Domain & Range

The domain and range depend on which direction the graph opens. Once you know a, b, h, k:

Condition	Domain	Range
b > 0 (goes right)	x ∈ [h, +∞)	Depends on a (see below)
b < 0 (goes left)	x ∈ (−∞, h]	Depends on a (see below)
a > 0 (goes up)	(above)	y ∈ [k, +∞)
a < 0 (goes down)	(above)	y ∈ (−∞, k]

✏️

Example: y = −9√(x − 2) + 4 → a = −9, b = 1, h = 2, k = 4
Goes down and right from vertex (2, 4)
Domain: x ∈ [2, +∞) · Range: y ∈ (−∞, 4]

4 Graphing Step-by-Step

Step 1: Rewrite in standard form y = a√(b(x − h)) + k — factor out the coefficient of x under the radical.
Step 2: Identify a, b, h, k.
Step 3: Plot the vertex (h, k).
Step 4: Determine direction (up/down, left/right) from signs of a and b.
Step 5: If graph heads toward origin, find the y-intercept (let x = 0, solve for y).
Step 6: Plot one or two extra points, connect smoothly.

⚠️

To factor: √(9x − 18) = √(9(x − 2)) = √9 · √(x − 2) = 3√(x − 2). The constant outside comes from pulling the coefficient out as a square root.

5 Finding Intercepts

Y-Intercept

Let x = 0, solve for y.
If the domain does not include 0, the function has NO y-intercept.

Y-intercept

Let x = 0 → solve for y

X-intercept (zero)

Let y = 0 → isolate the √, square both sides, solve

💡

When squaring both sides: (6/2)² = 9 → reverse squaring undoes the root. Always check that your answer satisfies the original equation — squaring can introduce extraneous solutions.

6 Finding the Rule

Given a vertex and one other point, you can find the full equation.

Step 1: Read vertex → gives h and k.
Step 2: Sketch to determine sign of b (left or right).
Step 3: Substitute the second known point (x, y) into y = a√(b(x − h)) + k.
Step 4: Solve for a.
Step 5: Write the complete rule.

✏️

Example: Vertex (−4, 4), y-intercept 10.
Sketch → goes right → b = +1.
10 = a√(1(0 − (−4))) + 4 → 6 = a√4 = 2a → a = 3.
Rule: y = 3√(x + 4) + 4

7 Solving Inequalities with Square Root Functions

To find the interval where f(x) ≥ c (or ≤ c):

Step 1: Set y = c and solve for x → find the point of intersection.
Step 2: Sketch both the function and the horizontal line y = c.
Step 3: Identify from the graph which x-interval satisfies the inequality.
Step 4: Write the answer using interval notation.

8 Common Mistakes

Mistake	Fix
Forgetting to factor before reading parameters	Always write as a√(b(x−h))+k first — never read from unfactored form.
Wrong vertex sign for h	In (x − h), if you see (x + 2) then h = −2, not +2. The formula subtracts h.
Not checking domain for y-intercept	If h > 0, the domain starts at h, so x = 0 may not be in domain → no y-intercept.
Forgetting ± when squaring	Squaring removes the root but can create extraneous solutions. Always verify.
Sketching in wrong direction	Sign of a = up/down, sign of b = left/right. Check both before sketching.