Questions: Identify the graph of the following function.
y = √(x-4)
Transcript text: Identify the graph of the following function.
\[
y=\sqrt{x-4}
\]
Solution
Solution Steps
To identify the graph of the function \( y = \sqrt{x-4} \), we need to understand its key characteristics:
The function \( y = \sqrt{x-4} \) is defined for \( x \geq 4 \) because the square root function requires non-negative arguments.
The graph will start at the point (4, 0) and will increase as \( x \) increases.
The shape of the graph will be a half-parabola opening to the right.
Step 1: Understanding the Function
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 \( k = 0 \).
Step 2: Determine the Domain
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 \( [4, \infty) \).
Step 3: Determine the Range
Since the square root function outputs non-negative values, the range of \( y = \sqrt{x - 4} \) is:
\[
[0, \infty)
\]
Step 4: Identify Key Points
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: \( (8, 2) \)
Step 5: Sketch the Graph
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
The graph of the function \( y = \sqrt{x - 4} \) is a half-parabola starting at \( (4, 0) \) and extending to the right. The domain is \( [4, \infty) \) and the range is \( [0, \infty) \).