Math Problem Statement
Given the graph of y equals f left parenthesis x right parenthesis as shown below, sketch the graph of y equals open vertical bar f left parenthesis x right parenthesis close vertical bar.
Solution
To sketch the graph of based on the graph provided, follow these steps:
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Identify the Portions of the Graph Where :
- These are the parts of the graph where is already non-negative. For these portions, the graph of will look exactly the same as the graph of .
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Identify the Portions of the Graph Where :
- These are the parts where the graph of is below the x-axis. For these portions, the graph of will be the reflection of the graph of over the x-axis. This is because taking the absolute value of a negative function value makes it positive.
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Sketch the Reflected Portion:
- Wherever the original graph is below the x-axis (where ), reflect that portion of the graph above the x-axis.
Application to Your Graph:
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Left Segment (below x-axis): The part of the graph on the left where is negative (below the x-axis) will be reflected above the x-axis. The shape of the curve remains the same, but it will now be above the x-axis.
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Right Segment (above x-axis): The part of the graph on the right where is already positive (above the x-axis) will remain unchanged.
Sketch:
- Starting from the left side:
- If the graph starts below the x-axis, reflect this portion above the x-axis.
- Move rightward:
- As soon as the graph crosses the x-axis (where ), continue with the graph as it is (no changes), since here.
Let me know if you want a detailed step-by-step explanation of any part of this or if you have any further questions!
Follow-up Questions:
- How do you determine where is negative or positive in a graph?
- What happens to the turning points of the graph when you apply the absolute value?
- Can the graph of ever go below the x-axis?
- How would the graph change if it were instead?
- What would the graph of look like if had no negative values?
- How can you check the accuracy of your reflected graph?
- What role does symmetry play in reflecting graphs over the x-axis?
- How would you graph for a constant ?
Tip: Reflecting a graph over the x-axis is equivalent to negating the y-values. For absolute value transformations, always focus on flipping negative y-values to positive while keeping the positive values unchanged.
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Math Problem Analysis
Mathematical Concepts
Graph transformations
Absolute value function
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12
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