.The graph of f(x)=cos(x) is transformed to a new function, g(x) , by stretching it horizontally by a factor of 4 and shifting it 1 unit up.
What is the equation of the new function g(x) ?
To find the equation of the new function, g(x), which is obtained by stretching the graph of f(x) = cos(x) horizontally by a factor of 4 and shifting it 1 unit up, we need to apply the appropriate transformations to the original function.
Let's start by looking at the transformations one by one:
1. Stretching horizontally by a factor of 4:
To stretch the graph horizontally by a factor of 4, we need to multiply the x-values by 1/4. This stretches the graph in the x-direction, making it narrower.
2. Shifting the graph 1 unit up:
To shift the graph 1 unit up, we need to add 1 to the y-values. This moves the entire graph vertically upward by 1 unit.
Now, let's apply these transformations to the function f(x) = cos(x):
1. Horizontal stretching:
Multiply the x-values by 1/4. This gives us f(x) = cos(1/4 * x).
2. Vertical shifting:
Add 1 to the y-values. This gives us g(x) = cos(1/4 * x) + 1.
Therefore, the equation of the new function, g(x), after stretching the graph of f(x) = cos(x) horizontally by a factor of 4 and shifting it 1 unit up, is g(x) = cos(1/4 * x) + 1.
To stretch the graph of f(x) horizontally by a factor of 4, we need to multiply the x-coordinate of each point on the graph by 1/4.
The equation for the horizontal stretch of g(x) becomes: g(x) = cos(1/4 * x)
To shift the stretched graph 1 unit up, we add 1 to the function:
g(x) = cos(1/4 * x) + 1
So, the equation of the new function g(x) is g(x) = cos(1/4 * x) + 1.