Questions: Identify the graph of the following function. y = √(x-4)

Solution Steps

To identify the graph of the function \( y = \sqrt{x-4} \), we need to understand its key characteristics:

1. The function \( y = \sqrt{x-4} \) is defined for \( x \geq 4 \) because the square root function requires non-negative arguments.
2. The graph will start at the point (4, 0) and will increase as \( x \) increases.
3. The shape of the graph will be a half-parabola opening to the right.

The given function is: \[ y = \sqrt{x - 4} \]

This is a square root function, which typically has the form \( y = \sqrt{x - h} + k \). In this case, \( h = 4 \) and \( k = 0 \).

The domain of the function \( y = \sqrt{x - 4} \) is determined by the requirement that the expression under the square root must be non-negative: \[ x - 4 \geq 0 \] Solving for \( x \): \[ x \geq 4 \] Thus, the domain is \( [4, \infty) \).

Since the square root function outputs non-negative values, the range of \( y = \sqrt{x - 4} \) is: \[ [0, \infty) \]

To graph the function, identify key points by substituting values of \( x \) within the domain:

• When \( x = 4 \): \[ y = \sqrt{4 - 4} = \sqrt{0} = 0 \] Point: \( (4, 0) \)

• When \( x = 5 \): \[ y = \sqrt{5 - 4} = \sqrt{1} = 1 \] Point: \( (5, 1) \)

• When \( x = 8 \): \[ y = \sqrt{8 - 4} = \sqrt{4} = 2 \] Point: \( (8, 2) \)

The graph of \( y = \sqrt{x - 4} \) starts at the point \( (4, 0) \) and increases gradually to the right. It is a half-parabola opening to the right.

Final Answer