My textbook asks me to graph out two equations: x^2+y^2=25 and y=sqrt(-x^2+25)
I think that algebraically these are identical.
In the solutions, the text shows two different graphs. The text says that the first equation fails the vertical line test, because its graph is essentially a circle. However, it says that the second equation does not. Only the top half of the graph (the half over y=0) is shown, although there are arrows pointing at the two ends of the graph.
My question is, how could one graph pass the vertical line test and the other not? Aren't the two equations, and therefore the graphs, identical?
The first one is a circle. The next one is half of a circle.
Solving the first one for y, you will obtain TWO formulas. y=+sqrt(25-x^2) OR y=-sqrt(25-x^2). When solved for y in this form, y cannot be both at the same time.