what is the difference between the graphs of the radical functions when vertical and horizontal stretches and compressions are applied?
same as with any other functions.
f(kx) is a dilation of 1/k
kx grows k times a fast as x, so the graph is compressed by a factor of 1/k
When vertical and horizontal stretches and compressions are applied to radical functions, they affect the shape and size of the graphs. The main difference between the two is that vertical stretches/compressions affect the amplitude or height, while horizontal stretches/compressions affect the width or horizontal scaling.
To understand the effects of these transformations, let's consider the square root function, f(x) = √x, as an example.
1. Vertical Stretches/Compressions:
A vertical stretch or compression is achieved by multiplying the function by a constant, typically denoted as "a" outside the square root symbol. For example, if we consider the function g(x) = a√x, where a > 1 represents a stretch and 0 < a < 1 represents a compression, we can observe the following:
- Stretch: As "a" increases, the graph of the function g(x) becomes steeper and taller. The points on the graph get further away from the x-axis, making the curve more stretched vertically.
- Compression: As "a" decreases, the graph of the function g(x) becomes shorter and wider. The points on the graph get closer to the x-axis, making the curve more compressed vertically.
2. Horizontal Stretches/Compressions:
A horizontal stretch or compression is achieved by manipulating the x-values inside the square root symbol. For example, if we consider the function h(x) = √(bx), where b > 1 represents a compression and 0 < b < 1 represents a stretch, we can observe the following:
- Compression: As "b" increases, the graph of the function h(x) becomes narrower. The points on the graph shift towards the y-axis, making the curve more compressed horizontally.
- Stretch: As "b" decreases, the graph of the function h(x) becomes wider. The points on the graph move away from the y-axis, making the curve more stretched horizontally.
It's important to remember that these transformations affect the shape of the graph, but the basic characteristics, such as the vertex (for square root functions), remain the same. Additionally, combinations of vertical and horizontal stretches and compressions can also be applied simultaneously to produce more complex transformations.