The graph of a function is horizontally compressed by a factor of 5 and vertically compressed by a factor of 2 Find an equation for this compressed function in terms of the function f(x)
The equation for the compressed function is g(x) = (1/2)f(5x).
To horizontally compress a function by a factor of 5, we need to divide the input (x-values) of the original function by 5. To vertically compress a function by a factor of 2, we need to multiply the output (y-values) of the original function by 2.
Let's assume the equation of the original function is f(x). To find the equation for the compressed function, we can write it as:
g(x) = 2 * f(x/5)
In this equation, g(x) represents the compressed function. We divide the input (x) by 5 to horizontally compress the function, and then multiply the output (f(x)) by 2 to vertically compress the function.
To find the equation for the horizontally and vertically compressed function, we need to understand how these types of transformations affect the original function.
If the original function is represented by f(x), then the horizontal compression by a factor of 5 would mean that each x-coordinate of the graph is divided by 5. This is because when we compress horizontally, we want to squeeze the graph towards the y-axis, making it narrower. To achieve this, we need to reduce the x-values.
Vertical compression by a factor of 2 means that each y-coordinate is multiplied by 2. This is because when we compress vertically, we want to squeeze the graph towards the x-axis, making it shorter. To achieve this, we need to stretch or multiply the y-values.
Combining these two transformations, the equation of the compressed function (let's call it g(x)) in terms of the original function f(x) can be written as:
g(x) = 2 * f(x/5)
Here, we divided x by 5 to account for horizontal compression and multiplied f(x/5) by 2 to account for vertical compression.
Therefore, g(x) = 2 * f(x/5) is the equation for the compressed function.