which graph represents $y = \\sqrt{x}$?
which graph represents $y = \\sqrt{x}$?
Answer
Explanation:
Step1: Analyze the function ( y = \sqrt{x} )
The domain of ( y=\sqrt{x} ) is ( x\geq0 ) (since we can't take the square root of a negative number in real numbers), and the range is ( y\geq0 ) (because the square root of a non - negative number is non - negative). So the graph should only exist in the first quadrant (where ( x\geq0 ) and ( y\geq0 )) and start at the origin ((0,0)).
Step2: Analyze each graph
- For the first graph (top - most): The curve is in the fourth quadrant (where ( y<0 )) for some part, which does not satisfy ( y = \sqrt{x}) (since ( y\geq0 ) for ( y=\sqrt{x})).
- For the second graph (second from top): The curve starts at the origin ((0,0)) and is in the first quadrant ( ( x\geq0 ) and ( y\geq0 )) as ( x ) increases, ( y=\sqrt{x}) also increases (for example, when ( x = 0,y = 0); when ( x=4,y = 2); when ( x = 9,y = 3)), and the shape of the curve matches the square - root function.
- For the third graph (third from top): The curve is in the second quadrant (where ( x<0 )) for some part, which does not satisfy the domain ( x\geq0 ) of ( y=\sqrt{x}).
- For the fourth graph (bottom - most): The curve is in the fourth quadrant (where ( y<0 )) for some part, which does not satisfy ( y\geq0 ) of ( y=\sqrt{x}).
Answer:
The second graph (the one in the middle among the four graphs, with the curve starting at the origin and lying in the first quadrant)