What are the coordinates of the minimum point of s(x)=x+4−−−−√
?
A.(4, 0)

B.(0, 4)

C.(−4, 0)

D.(0, −4)


To find the coordinates of the minimum point of the function \( s(x) = \sqrt{x + 4} \), we can follow these steps:

### 1. **Domain of the Function:**
The function \( s(x) = \sqrt{x + 4} \) involves a square root, which means the expression inside the square root must be non-negative (greater than or equal to zero) for the function to be defined. 

\[
x + 4 \geq 0
\]

\[
x \geq -4
\]

Thus, the domain of the function is \( x \geq -4 \).

### 2. **Finding the Minimum Point:**
Since the square root function \( \sqrt{x + 4} \) is defined only for \( x \geq -4 \) and the square root function is increasing (its derivative is positive for all \( x \) in its domain), the minimum value occurs at the leftmost point in its domain. This is \( x = -4 \).

Substituting \( x = -4 \) into the function:

\[
s(-4) = \sqrt{-4 + 4} = \sqrt{0} = 0
\]

### 3. **Coordinates of the Minimum Point:**
The minimum point occurs at \( x = -4 \) and \( s(-4) = 0 \). Thus, the coordinates of the minimum point are:

\[
(-4, 0)
\]

### Answer:
The correct answer is:

**C. \( (-4, 0) \)**

Would you like more details or have any questions about this process?

**Here are 5 related questions to expand understanding:**

1. How would the minimum point change if the function were \( s(x) = \sqrt{x + c} \), where \( c \) is a different constant?
2. What is the effect of squaring both sides on the domain and range of a function involving square roots?
3. If we defined a function \( s(x) = \sqrt{x + 4} + k \), where \( k \) is a constant, how would the graph shift?
4. How can we determine the range of the function \( s(x) = \sqrt{x + 4} \)?
5. How would the graph of \( s(x) = \sqrt{x + 4} \) change if it were \( s(x) = -\sqrt{x + 4} \)?

**Tip:** For functions involving square roots, always check the domain to ensure the argument of the square root is non-negative. This will help you identify where the function is defined and possible minimum or maximum points.

Learn how to find the coordinates of the minimum point for the function s(x) = sqrt(x + 4). Detailed steps include identifying the domain, determining where the function is defined, and calculating the minimum point. Suitable for high school students in Grades 10-12.

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