Understanding the Square Root Parent Function: Graph, Domain, Range, and Properties

Square Root Parent Function

The square root function, often denoted as f(x) = √x, is called a parent function because it serves as the base or standard form of the square root function

The square root function, often denoted as f(x) = √x, is called a parent function because it serves as the base or standard form of the square root function. The square root function is an example of a radical function, which involves taking the square root of a number.

The graph of the square root parent function, y = √x, is a curve that starts at the point (0, 0) and extends to the positive x-axis. As x increases, the y-value (or the square root of x) also increases, but at a decreasing rate.

Here are some key properties of the square root parent function:

1. Domain: The domain of the square root parent function is all non-negative real numbers, since the square root of a negative number is undefined in the real number system.
Domain: x ≥ 0

2. Range: The range of the square root parent function is all non-negative real numbers, since the square root of a non-negative number always produces a non-negative result.
Range: y ≥ 0

3. Symmetry: The square root parent function is symmetric about the y-axis. This means that if you reflect any point (x, y) on the graph across the y-axis, you will get the point (-x, y).

4. Increasing and decreasing: The square root parent function is continuously increasing, but at a decreasing rate. As x increases, y also increases, but the rate at which y increases gets slower as x becomes larger.

5. Vertical asymptote: The graph of the square root parent function approaches the x-axis (y = 0) but does not touch it. This creates a vertical asymptote at x = 0.

To graph the square root parent function, you can choose some x-values, calculate the corresponding y-values by taking the square root, and plot the points on a coordinate plane. Since the function is continuously increasing, you can connect the points smoothly to create the graph.

For example, if you choose x = 4, the corresponding y-value is y = √4 = 2. So the point (4, 2) would be on the graph. Similarly, if you choose x = 9, the corresponding y-value is y = √9 = 3. So the point (9, 3) would also be on the graph.

By plotting enough points and connecting them, you would see that the graph of the square root parent function resembles a curve that starts at the origin (0, 0) and extends to the positive x-axis.

Remember that any transformations to the square root parent function, such as shifting it vertically, horizontally, or stretching it, can change the shape and position of the graph. The square root parent function serves as a reference or starting point for understanding these transformations.

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