i don't understand what a horizontal or vertical stretching is of a graph.
y=x^3
Find graph horizontally stretched by a factor of 4
and vertically stretched by a factor of 4
y
=
x
^
3
To understand horizontal and vertical stretching, let's first look at the original function:
y = x^3
1. Horizontal Stretching:
A horizontal stretching of a graph occurs when the x-values of the original function are multiplied by a certain factor. In this case, we want to horizontally stretch the graph by a factor of 4.
To horizontally stretch the graph, divide the x-values by the stretching factor. So, to stretch the graph horizontally by a factor of 4, we can rewrite the function as:
y = (1/4)x^3
By dividing the x-values by 4, the graph of the function will become wider horizontally.
2. Vertical Stretching:
A vertical stretching of a graph occurs when the y-values of the original function are multiplied by a certain factor. In this case, we want to vertically stretch the graph by a factor of 4.
To vertically stretch the graph, multiply the y-values by the stretching factor. So, to stretch the graph vertically by a factor of 4, we can rewrite the function as:
y = 4x^3
By multiplying the y-values by 4, the graph of the function will become taller vertically.
To summarize, if the original function is y = x^3:
- A horizontal stretching by a factor of 4 will result in y = (1/4)x^3
- A vertical stretching by a factor of 4 will result in y = 4x^3
To understand horizontal and vertical stretching of a graph, it's important to first understand the concept of stretching. In mathematics, stretching refers to the modification of the shape, size, or dimensions of an object while maintaining its proportions.
In the context of graphing functions, a horizontal stretch or compression stretches or compresses the graph horizontally, altering its width. Conversely, a vertical stretch or compression stretches or compresses the graph vertically, changing its height.
Now let's tackle your specific example:
Given the equation y = x^3, to find a graph horizontally stretched by a factor of 4, we need to modify the x-values.
1. Multiply the x-values by the desired factor of stretching (4 in this case). This means multiplying each x-value by 4.
For example, if the original x-value was 2, after the horizontal stretching it becomes 2 * 4 = 8.
The new equation will be y = (4x)^3.
On the other hand, for a vertical stretch by a factor of 4, we modify the y-values instead.
1. Multiply the y-values by the desired factor of stretching (4 in this case). This means multiplying each y-value by 4.
For example, if the original y-value was 5, after the vertical stretching it becomes 5 * 4 = 20.
The new equation will be y = 4x^3.
To better visualize the effects of the horizontal and vertical stretches, it is recommended to plot the original equation (y = x^3) and the modified equations (y = (4x)^3 and y = 4x^3) on a coordinate plane or using graphing software.
Remember, horizontal stretching changes the x-values and vertical stretching changes the y-values.