Let's start by recalling the general form of an absolute value function.
f(x)= a|x- h|+ k
In this form, the vertex of the graph of the function is the point ( h, k). Now, consider the given function.
f(x)=a|bx-1|+c
We want to find the vertex of the graph of this function. To do so, let's rewrite it in the general form.
We are now able to identify the vertex of the graph of this function.
f(x)= a|b||x- 1/b|+ c
The vertex is the point ( 1b,c). This means that the graph of f(x) changes direction at the point ( 1b,c).
Alternative Solution
Finding the Coordinates of the Vertex Separately
Once again, consider the given absolute value function.
f(x)=a|bx-1|+c
The vertex of an absolute value function is the point at which its graph changes direction. Therefore, another way to find the vertex of the graph is by considering the cases of when the function will change directions.
Case I: If a is a negative number, then all possible y-values will be less than c. In this case, the maximum point will be the point where the expression inside the absolute value is 0 and y=c.
Case II: If a is a positive number, then all possible y-values will be greater than c. In this case, the minimum point will be the point where the expression inside the absolute value is 0 and y=c.
Note that in both cases, we can find the x-value of the vertex by setting the expression inside the absolute value, bx-1, equal to 0.
