Answer :
To determine the characteristics of the graph of the square root parent function, we can analyze each of the given statements:
### A. It starts at the origin.
- The square root function is defined as [tex]\( y = \sqrt{x} \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = \sqrt{0} = 0 \)[/tex].
Thus, the graph of the square root function starts at the origin (0,0). Therefore, statement A is True.
### B. It has a range of [tex]\( y \geq 0 \)[/tex].
- The output of the square root function, [tex]\( \sqrt{x} \)[/tex], is always zero or positive because the square root of a non-negative number is always non-negative.
This means the range of the square root function is [tex]\( y \geq 0 \)[/tex]. Therefore, statement B is True.
### C. It is a straight line.
- The graph of [tex]\( y = \sqrt{x} \)[/tex] is not a straight line; it is a curve. A straight line has a constant rate of change, whereas the rate of change of [tex]\( y = \sqrt{x} \)[/tex] decreases as [tex]\( x \)[/tex] increases.
Therefore, statement C is False.
### D. It has a domain of [tex]\( x \geq 0 \)[/tex].
- The square root function [tex]\( y = \sqrt{x} \)[/tex] is defined for all non-negative values of [tex]\( x \)[/tex], meaning [tex]\( x \geq 0 \)[/tex].
Thus, the domain of the square root function is [tex]\( x \geq 0 \)[/tex]. Therefore, statement D is True.
### Conclusion
Based on the analysis of each statement, the characteristics of the graph of the square root parent function are:
- A. It starts at the origin. (True)
- B. It has a range of [tex]\( y \geq 0 \)[/tex]. (True)
- C. It is a straight line. (False)
- D. It has a domain of [tex]\( x \geq 0 \)[/tex]. (True)
The correct answers are A, B, and D.
### A. It starts at the origin.
- The square root function is defined as [tex]\( y = \sqrt{x} \)[/tex].
- When [tex]\( x = 0 \)[/tex], [tex]\( y = \sqrt{0} = 0 \)[/tex].
Thus, the graph of the square root function starts at the origin (0,0). Therefore, statement A is True.
### B. It has a range of [tex]\( y \geq 0 \)[/tex].
- The output of the square root function, [tex]\( \sqrt{x} \)[/tex], is always zero or positive because the square root of a non-negative number is always non-negative.
This means the range of the square root function is [tex]\( y \geq 0 \)[/tex]. Therefore, statement B is True.
### C. It is a straight line.
- The graph of [tex]\( y = \sqrt{x} \)[/tex] is not a straight line; it is a curve. A straight line has a constant rate of change, whereas the rate of change of [tex]\( y = \sqrt{x} \)[/tex] decreases as [tex]\( x \)[/tex] increases.
Therefore, statement C is False.
### D. It has a domain of [tex]\( x \geq 0 \)[/tex].
- The square root function [tex]\( y = \sqrt{x} \)[/tex] is defined for all non-negative values of [tex]\( x \)[/tex], meaning [tex]\( x \geq 0 \)[/tex].
Thus, the domain of the square root function is [tex]\( x \geq 0 \)[/tex]. Therefore, statement D is True.
### Conclusion
Based on the analysis of each statement, the characteristics of the graph of the square root parent function are:
- A. It starts at the origin. (True)
- B. It has a range of [tex]\( y \geq 0 \)[/tex]. (True)
- C. It is a straight line. (False)
- D. It has a domain of [tex]\( x \geq 0 \)[/tex]. (True)
The correct answers are A, B, and D.
