Michael S. answered • 10/29/19
Mathematics Doctor and Instructor for All Levels
A point of clarification: Is the function y = -(sqrt(x+3)) + 1 or y = -(sqrt(x)+3) + 1? I would assume it is the former, but I will also provide an answer in case it is the latter.
This is a standard transformation of graphs problem. If we examine the equation we are given, we can start with the basic function y = sqrt(x). This is a graph with whose shape you should hopefully already be familiar. Because of this, we can perform a sequence of transformations on its graph based on the information we see in the equation.
First we look at the transformation to go from sqrt(x) to sqrt(x+3). This transformation will shift the graph to the left by 3. It might seem confusing that you shift to the left despite it being a +3 instead of -3. The best way to think about it is that if you have have sqrt(x+3), then you need to input something 3 less than you would for sqrt(x) in order to get the same result. For example, if you input 0 into sqrt(x), you get 0. In order to get 0 from sqrt(x+3), you need to input something 3 less, which is -3.
To go from sqrt(x+3) to -sqrt(x+3), reflect across the x axis, so that the graph looks upside down relative to the graph of sqrt(x+3). Lastly, to go from -sqrt(x+3) to -sqrt(x+3) + 1, shift the graph upwards.
In short: take sqrt(x), shift left 3, reflect over the x axis, then shift up 1.
It's important to perform these transformations in that order. If you shift up 1 before reflecting across the x-axis, for example, you will get a different result. In general, you want to work "inside out" in a sense. Start with the transformation closest to the x and then work outwards.
Once you have the graph, it should be easy to see that the maximum value is 1 (at x = -3) and there is no minimum value.
If the function in question were -(sqrt(x)+3) + 1, then you should shift up by 3, reflect across the x axis, then shift up by 1. This creates a maximum value of -2 at x=0 and no minimum value
