Understanding Square Root Functions

Square root functions contain a radical where the independent variable is inside the radical (the radicand). The parent function for all square root functions is f$x$ = √x, which passes through the points (0,0) and (1,1).

When working with square root functions, remember that you can't take the square root of a negative number in the real number system. This means the domain of a square root function is restricted to values where the radicand is non-negative (x ≥ 0).

Let's look at an example: y = -4√x. To graph this, create a table of values starting with x = 0, then plot points and connect them with a smooth curve. The negative coefficient $-4$ means this function is a vertically stretched and reflected version of the parent function. Its domain is x ≥ 0, and its range is y ≤ 0.

💡 When you see a negative coefficient in front of a square root $like -4√x$, it means the function is flipped upside down compared to the parent function.

Transformations of Square Root Functions

When a square root function has additions or subtractions inside the radical, it shifts horizontally. For example, in y = √$x-2$, the graph shifts right by 2 units. Its domain becomes x ≥ 2 since the radicand must be non-negative.

More complex functions like f$x$ = 3√$x-1$+2 combine multiple transformations. The steps to graph these functions are:

1. Find the domain by setting the radicand ≥ 0
2. Create a table of values
3. Plot points and draw a smooth curve
4. Determine the range by examining the lowest or highest possible y-values

For f$x$ = 3√$x-1$+2, the domain is x ≥ 1 (solving x-1 ≥ 0). The coefficient 3 stretches the function vertically, and the +2 shifts it up. This gives a range of y ≥ 2.

🔑 Remember this pattern: For f$x$ = a√$x-h$+k, the domain is x ≥ h, and the function is shifted h units right and k units up from the parent function.