Answer:
g(x) = -3/5|x -6| +4
Step-by-step explanation:
Given the graph of an absolute value function, you want the equation for it.
Transformations
The desired form is ...
g(x) = a|x -h| +k
The parameters a, h, k represent different features of the transformation of the graph of the parent function f(x) = |x|.
Vertical scale factor
The parameter 'a' is the vertical scale factor. There are two features of this graph that tell you what the value is.
- the slope of the lines
- the direction of opening
The line on the left side of the vertex seems to intersect points (1, 1) and (6, 4). The slope of this line can be found using the slope formula:
m = (y2 -y1)/(x2 -x1)
m = (4 -1)/(6 -1) = 3/5
The graph opens downward, meaning that the parent absolute value function has been reflected across the x-axis. In other words, the sign of 'a' is negative. Its magnitude is the slope we just found, so ...
Translation
When a function is translated h units horizontally and k units upward, the transformed function becomes ...
g(x) = f(x -h) +k
We notice the vertex of the graph has moved from (0, 0) to (6, 4), so we have ...
Equation
Now that we know the values of the parameters in the equation, we can write it as ...
[tex]g(x)=a|x-h|+k\\\\\boxed{g(x)=-\dfrac{3}{5}|x-6|+4}[/tex]
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Additional comment
Finding the vertical scale factor will work differently for different kinds of functions. In general, you want to find some vertical feature of the original function that you can recognize in the transformed function.
Here, that "feature" is the fact that an absolute value function has a rise of 1 for each run of 1 (a slope of 1) to the right of the vertex. This graph has a "rise" of -3 for a run of 5 to the right of the vertex. Effectively, the graph of the original absolute value function has been scaled by a factor of -3/5.
