Lesson 114: Graphing Square-Root Functions

New Concept

A square-root function is a function that contains a square root of a variable.

What’s next

Next, you’ll master graphing these functions by learning how to shift and reflect the basic parent function, $y = \sqrt{x}$.

Property

A function that contains a square root of a variable. The parent function is $y = \sqrt{x}$.

Explanation

Think of the parent function, $y = \sqrt{x}$, as the 'original superhero' of these graphs. Since we can't take the square root of a negative number and get a real answer, the part under the root (the radicand) must always be zero or positive. That’s the most important rule of the game!

Examples

The parent function's graph starts at $(0,0)$ and includes points like $(1,1)$, $(4,2)$, and $(9,3)$.
The function $y = \sqrt{x} + 5$ is a transformation of the parent function.
The function $y = -\sqrt{x}$ is a reflection of the parent function.

Before we can graph a square-root function, we must know where it even exists! This example shows how to find the function's valid starting point, which is a key idea from this lesson.

Example Problem

Determine the domain of the function $y = 5\sqrt{\frac{x}{3} + 2} - 9$.

Step-by-Step

1. The domain is the set of all possible input values for $x$. For a square-root function to be a real number, the expression inside the radical (the radicand) cannot be negative.
2. We set the radicand to be greater than or equal to zero. The terms outside the radical, like the $5$ and $-9$, do not affect the domain.

3. To solve for $x$, first subtract $2$ from both sides of the inequality.

4. Next, multiply both sides by $3$ to isolate $x$.

5. The domain is the set of all real numbers greater than or equal to $-6$. This means the graph will start at $x = -6$ and extend to the right.

Property

The domain is the set of possible input values for a function. For a square-root function, the radicand cannot be negative.

Explanation

To find the domain, just take whatever is under the square root sign and make sure it’s not a party pooper (negative)! Set the expression to be 'greater than or equal to zero' ($\ge 0$) and solve for $x$. This tells you all the 'allowed' x-values that won't break the math.

Examples

For $y = \sqrt{x-5}$, solve $x-5 \ge 0$ to get the domain $x \ge 5$.
For $y = \sqrt{2x+8}$, solve $2x+8 \ge 0$, which simplifies to $2x \ge -8$, so the domain is $x \ge -4$.

Property

Vertical translation: The graph of $f(x) = \sqrt{x} + c$ moves up (if $c>0$) or down (if $c<0$). Horizontal translation: The graph of $f(x) = \sqrt{x-c}$ moves right (if $c>0$) or left (if $c<0$).

Explanation

Think of your graph as a video game character. Adding a number outside the root ($+c$) makes it jump up or down. But if the number is inside with $x$ (like $x-c$), it's a sneaky side-step! Remember, $\sqrt{x-c}$ moves right, while $\sqrt{x+c}$ moves left.

Examples

The graph of $y = \sqrt{x} + 5$ is the parent function shifted 5 units up.
The graph of $y = \sqrt{x-4}$ is the parent function shifted 4 units to the right.
The graph of $y = \sqrt{x+1}$ is the parent function shifted 1 unit to the left.