Answer: choice C) g(x) = -2^(x+8)+3
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Explanation:
To go from f(x) to g(x), we need to do two transformations:
Transformation 1: shift 8 units to the left
Transformation 2: shift 3 units up
To pull off Transformation 1, we will replace every x with x+8. What this does is shift the entire xy axis 8 units to the right giving the illusion that the graph has moved 8 units to the left (when in reality the curve hasn't shifted at all). So that explains why we have x+8 there
Side Note: it is NOT x-8 because that will shift the graph to the right 8 units. So it is NOT choice A. This is a trap/trick answer.
Now onto Transformation 2. To do this, we effectively add 3 to both sides of the f(x) function. So f(x) = -2^x becomes f(x)+3 = -2^x + 3
Applying Transformation 1 and Transformation 2 at the same time has us go from f(x) = -2^x to f(x+8)+3 = -2^(x+8)+3, or in other words,
g(x) = -2^(x+8)+3 which is why choice C is the answer.
GeoGebra confirms the answer. See the attached image.
Side Note: it looks like f(x) touches or lays on the x axis, but this isn't the case. Instead, the function curve gets closer and closer to the x axis but never touches. There is a horizontal asymptote here. The same applies for g(x) as well but the horizontal asymptote is the line y = 3. It has been shifted up 3 units.
