Question

Which graph accurately demonstrates the relationship between the functions f(x)=x−−√
and f(x)=x−−√+3
?(1 point)
Responses

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Answer 1

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.
Image with alt text: Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 6 to 6 in increments of 2.

Two curves are graphed on a coordinate plane. The x-axis ranges from negative 3 to 3 in increments of 1. The y-axis ranges from negative 4 to 4 in increments of 2.

Answer 2

To accurately describe the relationship between the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \):

1. The function \( f(x) = \sqrt{x} \) is the standard square root function, which starts at the origin (0,0) and increases slowly as \( x \) increases. The range of this function is \( y \geq 0 \).

2. The second function \( f(x) = \sqrt{x} + 3 \) is simply the square root function shifted vertically upward by 3 units. Therefore, the new function will start at the point (0, 3) and also increases slowly from there. The range of this function is \( y \geq 3 \).

From the descriptions, the characteristics of these two functions should show that:
- The curve for \( f(x) = \sqrt{x} \) begins at (0,0) and stays below the one for \( f(x) = \sqrt{x} + 3 \), which starts at (0,3).

The possible graphs need to reflect this upward shift of 3 units for the second graph. It should be noted that both functions only exist for \( x ≥ 0 \) since the square root is not defined for negative \( x \).

None of the graphs described appear to give the correct range for \( f(x) = \sqrt{x} \) starting at (0,0) or for \( f(x) = \sqrt{x} + 3 \) starting at (0,3). The approximate y-ranges are also significant.

Based on this understanding, if you need to choose a graph, look for one where:
- One curve (the lower one) starts at (0, 0) and shapes to the right.
- The other curve (the upper one) starts at (0, 3) and also shapes to the right, remaining entirely above the first curve.

If you provide the corresponding graphs, I can help you choose the correct one.

Answer 3

Given the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \):

1. **\( f(x) = \sqrt{x} \)**:
- Starts at (0, 0) when \( x = 0 \) and is defined for \( x \geq 0 \).
- As \( x \) increases, \( f(x) \) increases but remains non-negative.

2. **\( f(x) = \sqrt{x} + 3 \)**:
- Starts at (0, 3) when \( x = 0 \) and is also defined for \( x \geq 0 \).
- This function is always 3 units higher than the first function.

### Analyzing potential graphs:

- The first function \( f(x) = \sqrt{x} \) should be visible in the lower part of the graph starting from (0, 0).
- The second function \( f(x) = \sqrt{x} + 3 \) should be visible starting from (0, 3) and above the first curve.

Now, considering the provided graph options:

1. **First option**: Y-axis ranges from -4 to 4.
2. **Second option**: Y-axis ranges from -6 to 6.
3. **Third option**: Y-axis ranges from -6 to 6 (same as the second).
4. **Fourth option**: Y-axis ranges from -4 to 4 (same as the first).

### Selection:

Both functions start at \( y \geq 0 \), so negative values on the y-axis in the first two options (-4 and -6) may render them less relevant unless those values are employed distinctly for the representation of the curves.

If we're focused on showing that \( f(x) = \sqrt{x} + 3 \) starts at (0, 3) and remains above \( f(x) = \sqrt{x} \), we should prioritize:

- **The first option** (Y-axis from -4 to 4) or **the fourth option** (also Y-axis from -4 to 4) could both show sufficient detail for the graphs of \( \sqrt{x} \) and \( \sqrt{x} + 3 \), but it is important that the curves remain non-negative.

Therefore, if I had to choose only one option, either the **first or fourth option** could potentially display the proper relationship. However, the crucial difference lies in ensuring that both curves do indeed appear correctly above zero on the y-axis where they should be. Without visual inspection, I would suggest going with the one where you can confirm this non-negativity validly.