Question

How do you find a vertex by looking at an absolute value equation?

Answer 1

Hint: The general form of an absolute value equation is \[y=a|x-h|+k\]. The variable tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables and tell us how far the graph shifts horizontally and vertically.

Complete step by step answer:
Absolute value equations are equations where the variable is within an absolute value operator, like \[|x-3|=5\]. The absolute value of a number depends on the number's sign: if it is positive, it is equal to the number: \[|5|=5\]. If the number is negative, then the absolute value is its opposite: \[|-5|=5\]. So when we are dealing with a variable, we need to consider both cases.
In the given equation \[y=a|x-h|+k\]. x is a variable. Let m be the slope of the equation. slope \[m=a\]on the right side of the vertex that is \[x>h\]and slope \[m=-a\]on the left side of the vertex that is \[xAs per the given question, we have to find the vertex of an absolute value equation.
The vertex of an absolute value equation \[y=a|x-h|+k\] is \[(h,k)\].
We can find a vertex by looking at the equation that is \[(h,k)\].
For example, let us take an absolute value equation \[y=5|x-9|+3\].
The vertex of the above equation is \[(9,3)\].
Therefore, we can find vertex of an absolute value equation by looking at it that is \[(h,k)\] for \[y=a|x-h|+k\].

Note:
While solving these types of problems, we need to have enough knowledge of absolute value equations and their functions. While finding vertex check whether it is in the form \[y=a|x-h|+k\] or \[y=a|x-h|-k\]. Because the vertex changes for \[y=a|x-h|-k\] that is \[(h,-k)\]. We should avoid calculation mistakes to get the correct results.