compare the graphs of f (x) = - sq rt x and g (x) = - 3 sq rt x. which of the following features abbout the graphs is true ?

Answer 1

To compare the graphs of the functions \( f(x) = -\sqrt{x} \) and \( g(x) = -3\sqrt{x} \), let's analyze a few key features:

1. Shape of the Graphs:

   • Both graphs will be downward-opening curves because they are both negative square root functions.

2. Vertical Stretch:

   • The graph of \( g(x) \) is a vertical stretch of the graph of \( f(x) \). Specifically, \( g(x) \) is stretched by a factor of 3 compared to \( f(x) \). This means that for any given \( x \), the value of \( g(x) \) will be three times that of \( f(x) \).
   • Example: For \( x = 1 \):
     • \( f(1) = -\sqrt{1} = -1 \)
     • \( g(1) = -3\sqrt{1} = -3 \)

3. Intercepts:

   • Both functions intersect the y-axis at the same point: \( (0, 0) \). This means they have the same x-intercept.

4. Domain:

   • The domain for both functions is \( x \geq 0 \) (since the square root function is defined for non-negative values).

5. Range:

   • The range for \( f(x) \) is \( (-\infty, 0] \) and for \( g(x) \), it is \( (-\infty, -3] \). This highlights that \( g(x) \) achieves lower values than \( f(x) \), reflecting the vertical stretch.

6. Comparison of Values:

   • For all \( x \geq 0 \), \( g(x) \) is always less than \( f(x) \) since multiplying by -3 makes the outputs more negative.

7. Graph Behavior:

   • Both graphs approach the horizontal line \( y = 0 \) as \( x \) approaches infinity, but \( g(x) \) lies entirely below \( f(x) \) due to the vertical stretch.

In summary, the key features that are true about the graphs of \( f(x) \) and \( g(x) \) are:

• They have the same shape (both are downward opening).
• \( g(x) \) is a vertical stretch of \( f(x) \) by a factor of 3.
• They share the same x-intercept (0) but different ranges.