The same types of transformations that create new linear functions, also do the same for absolute value functions. They affect absolute value functions in the same way as well. However, since linear functions and absolute value functions have some significant differences, the transformations might look different graphically.
To begin, we'll analyze the given function rules. Adding 1 to the input of f, and then subtracting the output by 3, gives the function g. We can recognize the addition to the input as a horizontal translation, but in which direction? When the input of a function is increased by some number the graph is translated to the left. Thus, this is a translation to the left, by 1 unit.We've established that f(x+1) is a translation of f(x) to the left by 1 unit. Now, we can view
Thus, f has been translated 1 unit to the left and 3 units downward to become g. The y-intercept is where the graph crosses the y-axis, at which point the x-coordinate is 0. We'll find the y-coordinate by substituting x=0 into the rule of g(x).
To fully evaluate g(0), we first have to find f(1) by substituting x=1 into the rule of f(x).
We can now use this value to find the y-coordinate.
Thus, the y-intercept of g is (0,-2).
Below, the graphs of f, g, and h are shown. Find the rules of g and h, expressed as transformations of f.
We'll start by comparing the graphs of f and g. Their vertices are at the same point, and g is flatter than f. This flattening could either be seen as the graph being stretched horizontally, or shrunk vertically, but which is it? The y-intercept has been affected, which rules out the horizontal stretch. At the same time, the x-intercept has been preserved, which confirms the vertical shrink.
By comparing their function values at different x-values, we can find the factor by which it's been shrunk. For instance, choosing x=-3 and x=3 gives us the information we need.At x=-3, the function values are 4 and 2. At x=3, the function values are 2 and 1. Thus, we can conclude that every function value of g is half of that of f. This transformation is written algebraically as