Question

Find the function whose graph is the graph of \(y=\sqrt{x}\) but reflected about the \(y\) -axis.

Short answer

The function reflected about the \( y \)-axis is \( y = \sqrt{-x} \).

Step-by-step solution

- Understand the Original Function

The given function is \( y = \sqrt{x} \). This is a standard square root function, where the output (\( y \)) is the square root of the input (\( x \)).

- Reflect the Function About the y-Axis

When reflecting a function about the \( y \)-axis, each \( x \)-coordinate of the function is replaced with \( -x \). This means you need to replace \( x \) in the original function with \( -x \). The transformation for our function \( y = \sqrt{x} \) becomes \( y = \sqrt{-x} \).

- Simplify the Transformed Function

The new function is \( y = \sqrt{-x} \), which represents the reflection of the original function about the \( y \)-axis. Ensure that the domain of the new function is considered. Since we can only take the square root of non-negative numbers, \( -x \geq 0 \) must hold, implying \( x \leq 0 \).

Key concepts

graph transformation

Graph transformation is a fundamental concept in understanding how functions change when their graphs undergo certain modifications.
There are various types of transformations, such as translations, reflections, and scaling.
In this exercise, we focus on the reflection of the function about the y-axis.

When reflecting a function about the y-axis, each x-coordinate in the function is replaced with its negative counterpart (-x).
This means that if you have a point \( (x, y) \) on the original graph, its image on the reflected graph will be \( (-x, y) \).

For example, for the function \( y = \sqrt{x} \), reflecting it about the y-axis would transform it to \( y = \sqrt{-x} \).
Essentially, this turns the graph around the y-axis, taking every point on the right side and placing it symmetrically on the left side.

square root function

The square root function is a common function in algebra, represented as \( y = \sqrt{x} \).
It is defined for all non-negative real numbers (x >= 0), producing a non-negative output (y >= 0) as well.
The graph of \( y = \sqrt{x} \) starts at the origin (0,0) and increases gradually, forming a curve that gets less steep as x increases.

Key features of the square root function include:

• The domain: x must be greater than or equal to 0.
• The range: y must be greater than or equal to 0.
• The function increases gradually and continuously but never decreases.
• The function passes through the point (1,1) because the square root of 1 is 1.

This function is often used to model various phenomena in science and engineering where the relationship between two variables exhibits a square root pattern.

domain of a function

The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Understanding the domain is essential because it tells us where the function is valid and can be calculated.
For the square root function, such as \( y = \sqrt{-x} \), the domain is restricted to values that make the expression inside the square root non-negative.

In the given solution, the new function after reflection is \( y = \sqrt{-x} \).
To find its domain, we need to ensure that the value inside the square root, -x, is non-negative, i.e., \[ -x \geq 0 \] or \[ x \leq 0 \].
Therefore, the domain of \( y = \sqrt{-x} \) is all non-positive real numbers (x ≤ 0).

Remember, the domain usually changes whenever we transform the graph or modify the function rules, so always check the new domain limits after performing operations.