How do you recognize the graph of y = -sqrt(x)?

You recognise the graph of \( y = -\sqrt{x} \) as a downward curve starting from the origin in the first quadrant.

To understand this better, let's break it down. The function \( y = -\sqrt{x} \) involves taking the square root of \( x \) and then applying a negative sign to the result. The square root function, \( \sqrt{x} \), is only defined for \( x \geq 0 \) because you can't take the square root of a negative number in real numbers. This means the graph will only exist for non-negative values of \( x \).

The basic shape of \( \sqrt{x} \) is a curve that starts at the origin (0,0) and rises slowly to the right. When you apply the negative sign, it flips this curve upside down. So, instead of rising, the curve will fall as you move to the right. This gives you a downward curve that starts at the origin and moves into the first quadrant.

In terms of coordinates, for any positive \( x \), the corresponding \( y \) value will be negative. For example, if \( x = 1 \), then \( y = -\sqrt{1} = -1 \). If \( x = 4 \), then \( y = -\sqrt{4} = -2 \). This pattern continues, showing that as \( x \) increases, \( y \) becomes more negative, but the rate of decrease slows down because the square root function grows slower as \( x \) gets larger.

So, when you plot \( y = -\sqrt{x} \), you get a curve that starts at the origin and gently slopes downward to the right, staying entirely in the first quadrant of the coordinate plane.