1. Absolute Value Functions
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Think about how variables in those positions affect the graphs of linear functions.
a can be a reflection, stretch, or shrink.
h is a horizontal translation.
k is a vertical translation.
Transformations of absolute value functions are very similar to transformations of linear functions, with the main difference being that each transformation must be considered on both halves of the V-shape. Let's look at the given function. g(x)=a|x-h|+k We can compare how each of these variables affect the parent function f(x)=|x|.
If we have f(x)=|x| and g(x)=a|x|, there are a few things that a can do. First, if a is negative, no matter how small or large a number, the graph is going to be a reflection. Every output will have its sign changed.
If a>1 or a<-1, then the V-shape of the function is going to become thinner. This is a vertical stretch because it is stretching away from the x-axis.
If -1
If f(x)=|x| and g(x)=|x-h|, this means that each value of x is shifted before the absolute value is calculated. This is a horizontal translation and can be in the positive or the negative direction.
If f(x)=|x| and g(x)=|x|+k, this means that each value of y is shifted after the absolute value is calculated. This is a vertical translation and can be in the positive or the negative direction.
