Question

identify the graph of the function \\(y = -3\sqrt{x}\\).

Explanation:

Analyze the parent function

The given function is \(y = -3\sqrt{x}\). The parent function is the Square Root Function, \(f(x) = \sqrt{x}\), which starts at the origin \((0,0)\) and curves upwards into the first quadrant.

Apply function transformations

Using Function Transformations, we analyze the modifications:

1. The negative sign in front of the coefficient \(-3\) represents a reflection across the x-axis. This means the graph must curve downwards into the fourth quadrant.
2. The factor of \(3\) represents a Vertical Stretch by a factor of 3.

Evaluate key points

We calculate specific coordinate points to identify the correct graph:

• For \(x = 0\): \(y = -3\sqrt{0} = 0\). The starting point is \((0,0)\).
• For \(x = 1\): \(y = -3\sqrt{1} = -3\).
• For \(x = 4\): \(y = -3\sqrt{4} = -3(2) = -6\).
• For \(x = 9\): \(y = -3\sqrt{9} = -3(3) = -9\).

Match with the given options

We check the coordinates on the provided graphs:

• Graph a: Starts at \((0,0)\). At \(x = 1\), \(y = -3\). At \(x = 4\), \(y = -6\). At \(x = 9\), \(y = -9\). This perfectly matches our calculated points.
• Graph b: Curves upwards, which represents \(y = 3\sqrt{x}\) (no reflection).
• Graph c: At \(x = 1\), \(y \approx -1.5\), and at \(x = 4\), \(y \approx -3\). This represents a vertical compression, specifically \(y = -1.5\sqrt{x}\).

Answer:

• a. The graph starting at (0,0) and passing through (1,-3), (4,-6), and (9,-9) (Correct answer)
• b. The graph starting at (0,0) and curving upwards through (1,3), (4,6), and (9,9)
• c. The graph starting at (0,0) and passing through (1,-1.5), (4,-3), and (9,-4.5)