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Which of the following functions represents the graph of f(x)=√x+1-2? 6 5 4 3 2 f(x) -10 6 7 -1 O O f(x) -T -4 -5 4 -3 -2 2 -4 - 6 5 4 3 2 1 °577? -1 -2 -3 2 3 6 7

Which of the following functions represents the graph of f(x)=√x+1-2? 6 5 4 3 2 f(x) -10 6 7 -1 O O f(x) -T -4 -5 4 -3 -2 2 -4 - 6 5 4 3 2 1 °577? -1 -2 -3 2 3 6 7

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
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**Question:** Which of the following functions represents the graph of \( f(x) = \sqrt[3]{x-1} - 2 \)? **Graphs:** *First Graph:* The first graph depicts a function \( f(x) \) on a coordinate plane. The function appears as a curve with the following characteristics: - It starts in the third quadrant, rising from the lower left. - It approaches the y-axis as it rises. - Once it reaches near the x-axis, it starts to descend gradually and enters the fourth quadrant. - The curve is smooth and continuous through these quadrants. *Second Graph:* The second graph also depicts a function \( f(x) \) on a coordinate plane. This graph has the following features: - It starts in the third quadrant and rises sharply as it approaches the y-axis. - It reaches the x-axis and levels off, starting to descend gradually. - The function moves into the fourth quadrant, continuing smoothly. ### Graph Analysis Both graphs appear similar in nature, showing the general traits of a cube root function, which is modified by a horizontal shift (to the right by 1 unit) and a vertical shift (downward by 2 units). The correct graph will match the given function \( f(x) = \sqrt[3]{x-1} - 2 \). To determine which graph accurately represents this function: - Consider the transformations applied to the basic cube root function. - The rightward shift by 1 unit and downward shift by 2 units result in: - The point (0, 0) of the standard cube root function \(\sqrt[3]{x} \) moving to (1, -2). Check for this characteristic in the given options. **Conclusion:** Select the graph that precisely aligns with these transformations and the described behavior of the function.
Transcribed Image Text:**Question:** Which of the following functions represents the graph of \( f(x) = \sqrt[3]{x-1} - 2 \)? **Graphs:** *First Graph:* The first graph depicts a function \( f(x) \) on a coordinate plane. The function appears as a curve with the following characteristics: - It starts in the third quadrant, rising from the lower left. - It approaches the y-axis as it rises. - Once it reaches near the x-axis, it starts to descend gradually and enters the fourth quadrant. - The curve is smooth and continuous through these quadrants. *Second Graph:* The second graph also depicts a function \( f(x) \) on a coordinate plane. This graph has the following features: - It starts in the third quadrant and rises sharply as it approaches the y-axis. - It reaches the x-axis and levels off, starting to descend gradually. - The function moves into the fourth quadrant, continuing smoothly. ### Graph Analysis Both graphs appear similar in nature, showing the general traits of a cube root function, which is modified by a horizontal shift (to the right by 1 unit) and a vertical shift (downward by 2 units). The correct graph will match the given function \( f(x) = \sqrt[3]{x-1} - 2 \). To determine which graph accurately represents this function: - Consider the transformations applied to the basic cube root function. - The rightward shift by 1 unit and downward shift by 2 units result in: - The point (0, 0) of the standard cube root function \(\sqrt[3]{x} \) moving to (1, -2). Check for this characteristic in the given options. **Conclusion:** Select the graph that precisely aligns with these transformations and the described behavior of the function.
### Graph Analysis for Educational Purposes #### Graph Descriptions **Top Graph:** - The graph represents a function \( f(x) \). - The x-axis ranges from -5 to 7. - The y-axis ranges from -7 to 7. - The function \( f(x) \) appears to be decreasing as \( x \) increases. - Around \( x = -2 \), the function shows a sharp decrease in its slope, indicating a point of rapid change. - Beyond \( x = -1 \), the function continues to decrease but at a lesser slope, gradually approaching a negative y-value. **Bottom Graph:** - Similar to the top graph, this graph also represents the function \( f(x) \). - The ranges for both axes (x-axis: -5 to 7, y-axis: -7 to 7) are the same. - The function’s behavior and general shape are identical to the top graph, indicating that they are likely the same function. ### Detailed Explanation Both graphs depict the same function \( f(x) \) plotted over the same range for both the x and y axes. The function starts off relatively flat and close to the x-axis but quickly drops at \( x = -2 \), suggesting a critical point or a steep slope. Post \( x = -2 \), the function continues to decrease but more gradually. These graphs serve as visual aids for understanding how functions behave—specifically, how they change their slope and direction as \( x \) varies. This type of visual representation is crucial for students learning about calculus and the behavior of differentiable functions. In a classroom setting, instructors can use these graphs to: - Discuss critical points and points of inflection. - Explain the concept of decreasing functions. - Show real-life applications of function behavior over different intervals. Remember, interpreting graphs is essential for a deeper understanding of mathematics and its broader applications in various fields such as physics, engineering, and economics.
Transcribed Image Text:### Graph Analysis for Educational Purposes #### Graph Descriptions **Top Graph:** - The graph represents a function \( f(x) \). - The x-axis ranges from -5 to 7. - The y-axis ranges from -7 to 7. - The function \( f(x) \) appears to be decreasing as \( x \) increases. - Around \( x = -2 \), the function shows a sharp decrease in its slope, indicating a point of rapid change. - Beyond \( x = -1 \), the function continues to decrease but at a lesser slope, gradually approaching a negative y-value. **Bottom Graph:** - Similar to the top graph, this graph also represents the function \( f(x) \). - The ranges for both axes (x-axis: -5 to 7, y-axis: -7 to 7) are the same. - The function’s behavior and general shape are identical to the top graph, indicating that they are likely the same function. ### Detailed Explanation Both graphs depict the same function \( f(x) \) plotted over the same range for both the x and y axes. The function starts off relatively flat and close to the x-axis but quickly drops at \( x = -2 \), suggesting a critical point or a steep slope. Post \( x = -2 \), the function continues to decrease but more gradually. These graphs serve as visual aids for understanding how functions behave—specifically, how they change their slope and direction as \( x \) varies. This type of visual representation is crucial for students learning about calculus and the behavior of differentiable functions. In a classroom setting, instructors can use these graphs to: - Discuss critical points and points of inflection. - Explain the concept of decreasing functions. - Show real-life applications of function behavior over different intervals. Remember, interpreting graphs is essential for a deeper understanding of mathematics and its broader applications in various fields such as physics, engineering, and economics.
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