Question
Which graph accurately demonstrates the relationship between the functions f(x)=x−−√
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𝑥
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=
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and f(x)=x−−√+3
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)
=
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3
?(1 point)
Responses
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.
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.
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.
𝑓
(
𝑥
)
=
𝑥
and f(x)=x−−√+3
𝑓
(
𝑥
)
=
𝑥
+
3
?(1 point)
Responses
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.
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.
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.
Answers
Answered by
GPT-4o mini
To determine which graph accurately represents the functions \( f(x) = \sqrt{x} \) and \( f(x) = \sqrt{x} + 3 \), we need to consider the transformations between these two functions.
1. **Function Analysis**:
- The function \( f(x) = \sqrt{x} \) is the standard square root function. It starts at the point (0, 0) and increases as \( x \) increases.
- The function \( f(x) = \sqrt{x} + 3 \) is the same square root function, but it has been vertically shifted upwards by 3 units. Therefore, it starts at (0, 3) and also increases as \( x \) increases.
2. **Key Characteristics**:
- The first function, \( \sqrt{x} \), has points: (0, 0), (1, 1), (4, 2), etc.
- The second function, \( \sqrt{x} + 3 \), has points: (0, 3), (1, 4), (4, 5), etc.
3. **Graph Analysis**:
- In the graph representing both functions, look for the curve for \( \sqrt{x} \) starting at (0,0) and going upwards to the right.
- The second curve for \( \sqrt{x} + 3 \) should start at (0,3) and follow a parallel path above the first curve.
Based on that analysis, you should look for the graph where one of the curves starts at the origin (0,0) and another one starts at (0,3) and runs parallel to the first curve but above it.
If the images you have don't provide enough differentiation in what you see (assuming they are all similar), then look specifically for the curve starting higher up at (0,3) to represent \( \sqrt{x} + 3 \). The correct graph will show a clear vertical shift upwards between the two functions.
1. **Function Analysis**:
- The function \( f(x) = \sqrt{x} \) is the standard square root function. It starts at the point (0, 0) and increases as \( x \) increases.
- The function \( f(x) = \sqrt{x} + 3 \) is the same square root function, but it has been vertically shifted upwards by 3 units. Therefore, it starts at (0, 3) and also increases as \( x \) increases.
2. **Key Characteristics**:
- The first function, \( \sqrt{x} \), has points: (0, 0), (1, 1), (4, 2), etc.
- The second function, \( \sqrt{x} + 3 \), has points: (0, 3), (1, 4), (4, 5), etc.
3. **Graph Analysis**:
- In the graph representing both functions, look for the curve for \( \sqrt{x} \) starting at (0,0) and going upwards to the right.
- The second curve for \( \sqrt{x} + 3 \) should start at (0,3) and follow a parallel path above the first curve.
Based on that analysis, you should look for the graph where one of the curves starts at the origin (0,0) and another one starts at (0,3) and runs parallel to the first curve but above it.
If the images you have don't provide enough differentiation in what you see (assuming they are all similar), then look specifically for the curve starting higher up at (0,3) to represent \( \sqrt{x} + 3 \). The correct graph will show a clear vertical shift upwards between the two functions.