How is the graph of the parent function, y = StartRoot x EndRoot transformed to produce the graph of y = StartRoot negative 2 x EndRoot?
y=√x
reflection across the y-axis: y=√(-x)
scale horizontally by a factor of 1/2: y=√(-2x)
To transform the graph of the parent function y = √x to y = √(-2x), there are a few steps involved:
1. The negative sign outside the function indicates a reflection across the x-axis.
2. The coefficient -2 inside the function stretches the graph horizontally by a factor of 1/2. This means that every x-coordinate of the parent function is multiplied by 1/2 to obtain the corresponding x-coordinate of the new function.
Applying these transformations to the parent function, the graph of y = √(-2x) is a reflection across the x-axis and is horizontally compressed by a factor of 1/2 compared to the graph of y = √x.
To understand how the graph of the parent function, y = √x, is transformed to y = √(-2x), we need to analyze the effect of the transformation on the parent function.
The transformation from y = √x to y = √(-2x) involves two main steps: a reflection and a horizontal compression.
1. Reflection:
The negative sign in front of the "2" reflects the graph of y = √x about the y-axis. This means that all the y-values remain the same, but the x-values are reversed. For example, if the parent function has a point (2, 1), the transformed function will have a corresponding point (-2, 1).
2. Horizontal Compression:
The number "2" inside the square root represents a horizontal compression. It indicates that the graph is being compressed horizontally by a factor of 2 compared to the parent function. This means that for every unit increase in x, the corresponding y-value only increases by half a unit. As a result, the graph becomes steeper.
By combining the reflection and the horizontal compression, the graph of y = √(-2x) will have the following characteristics:
- It will be a reflection of the parent function, y = √x, across the y-axis.
- The graph will be compressed horizontally by a factor of 2 compared to the parent function.
- The graph will have a steeper slope.
To visualize the transformation, you can plot points on both the parent function, y = √x, and the transformed function, y = √(-2x), and observe the changes in the graph.