What would be the steps for finding the equation for the function f(x) based on the graph (the blue line) and what would the equation be? include steps that are universal to finding the equation of any possible graphed function, as i really don't even know where to begin and would like to be able to solve problems like this on my own one day. : )

  f(x) = -2|x-2| +4

Step-by-step explanation:

Step 1: recognize this as an inverted, scaled, and translated absolute value function.

Step 2: identify the scale factor as 2, because each section has a "rise" of 2 for each "run" of 1. The scale factor is -2 because the function is inverted (reflected across the x-axis).

Step 3: identify the vertex* as (2, 4).

Step 4: Use the scale factor and translation information to make the transformation ...   (let abs(x) represent the absolute value function)

  f(x) = (scale factor)×abs(x -horizontal shift) + (vertical shift)

  f(x) = -2|x -2| +4

_____

* The vertex is an identifiable point on the graph of this function. Knowing how the vertex is translated tells you how the whole function is translated. In general, you want to use an identifiable point (a turning point, point of symmetry, extreme point, whatever) when you're trying to determine the translation. For some functions, like a line, there is no specific identifiable point, so determining any specific translation is difficult or impossible.

_____

Comment on writing functions from graphs

One of the reasons for studying different functions is so you can learn to recognize their shape, and associate a formula and certain properties with that shape.

_____

The attached graph shows the parent absolute value function in blue and the transformed function f(x) in red.

the answer is 99.