Question
Which is the graph of y = 2 – square root of x?
On a coordinate plane, a curved line approaches the grid line at (3, 2.3), crosses the y-axis at (0, 1.5) and ends at (negative 2, 0).
On a coordinate plane, a curved line approaches the grid line at (negative 3, 2.3), crosses the y-axis at (0, 1.5) and ends at (2, 0).
On a coordinate plane, a curved line approaches the grid line at (3, 0.3), and ends at (0, 2).
On a coordinate plane, a curved line approaches the grid line at (3, negative 0.3) and ends at (0, negative 2).
On a coordinate plane, a curved line approaches the grid line at (3, 2.3), crosses the y-axis at (0, 1.5) and ends at (negative 2, 0).
On a coordinate plane, a curved line approaches the grid line at (negative 3, 2.3), crosses the y-axis at (0, 1.5) and ends at (2, 0).
On a coordinate plane, a curved line approaches the grid line at (3, 0.3), and ends at (0, 2).
On a coordinate plane, a curved line approaches the grid line at (3, negative 0.3) and ends at (0, negative 2).
Answers
GPT-4o mini
To analyze the graph of the function \( y = 2 - \sqrt{x} \), we can start by identifying some key characteristics:
1. **Domain**: The square root function \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of the function \( y = 2 - \sqrt{x} \) is \( x \geq 0 \).
2. **Intercepts**:
- **Y-intercept**: When \( x = 0 \),
\[
y = 2 - \sqrt{0} = 2 - 0 = 2.
\]
So, the graph crosses the y-axis at (0, 2).
- **X-intercept**: Set \( y = 0 \):
\[
0 = 2 - \sqrt{x} \implies \sqrt{x} = 2 \implies x = 4.
\]
So, the graph crosses the x-axis at (4, 0).
3. **Behavior**: As \( x \) increases, \( y \) will decrease since \( \sqrt{x} \) increases. The function will approach \( y = 2 \) as \( x \) approaches 0, and it will decrease without bound as \( x \) increases beyond 4 (approaching negative values as \( x \) becomes larger).
Given these points, we look for a graph that:
- Crosses the y-axis at (0, 2).
- Crosses the x-axis at (4, 0).
- Exists only for \( x \geq 0 \).
None of the descriptions you provided approximate this behavior correctly, particularly none of them state that the graph crosses the y-axis at (0, 2) and the x-axis at (4, 0) while only existing in the first quadrant. Therefore, none of the options accurately represent the graph of \( y = 2 - \sqrt{x} \).
1. **Domain**: The square root function \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of the function \( y = 2 - \sqrt{x} \) is \( x \geq 0 \).
2. **Intercepts**:
- **Y-intercept**: When \( x = 0 \),
\[
y = 2 - \sqrt{0} = 2 - 0 = 2.
\]
So, the graph crosses the y-axis at (0, 2).
- **X-intercept**: Set \( y = 0 \):
\[
0 = 2 - \sqrt{x} \implies \sqrt{x} = 2 \implies x = 4.
\]
So, the graph crosses the x-axis at (4, 0).
3. **Behavior**: As \( x \) increases, \( y \) will decrease since \( \sqrt{x} \) increases. The function will approach \( y = 2 \) as \( x \) approaches 0, and it will decrease without bound as \( x \) increases beyond 4 (approaching negative values as \( x \) becomes larger).
Given these points, we look for a graph that:
- Crosses the y-axis at (0, 2).
- Crosses the x-axis at (4, 0).
- Exists only for \( x \geq 0 \).
None of the descriptions you provided approximate this behavior correctly, particularly none of them state that the graph crosses the y-axis at (0, 2) and the x-axis at (4, 0) while only existing in the first quadrant. Therefore, none of the options accurately represent the graph of \( y = 2 - \sqrt{x} \).