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 y=f(x)y = |f(x)| based on the graph provided, follow these steps:

  1. Identify the Portions of the Graph Where f(x)0f(x) \geq 0:

    • These are the parts of the graph where y=f(x)y = f(x) is already non-negative. For these portions, the graph of y=f(x)y = |f(x)| will look exactly the same as the graph of y=f(x)y = f(x).
  2. Identify the Portions of the Graph Where f(x)<0f(x) < 0:

    • These are the parts where the graph of y=f(x)y = f(x) is below the x-axis. For these portions, the graph of y=f(x)y = |f(x)| will be the reflection of the graph of y=f(x)y = f(x) over the x-axis. This is because taking the absolute value of a negative function value makes it positive.
  3. Sketch the Reflected Portion:

    • Wherever the original graph is below the x-axis (where f(x)<0f(x) < 0), reflect that portion of the graph above the x-axis.

Application to Your Graph:

  • Left Segment (below x-axis): The part of the graph on the left where f(x)f(x) 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.

  • Right Segment (above x-axis): The part of the graph on the right where f(x)f(x) 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 f(x)=0f(x) = 0), continue with the graph as it is (no changes), since f(x)0f(x) \geq 0 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:

  1. How do you determine where f(x)f(x) is negative or positive in a graph?
  2. What happens to the turning points of the graph when you apply the absolute value?
  3. Can the graph of y=f(x)y = |f(x)| ever go below the x-axis?
  4. How would the graph change if it were y=f(x)y = -f(x) instead?
  5. What would the graph of y=f(x)y = |f(x)| look like if f(x)f(x) had no negative values?
  6. How can you check the accuracy of your reflected graph?
  7. What role does symmetry play in reflecting graphs over the x-axis?
  8. How would you graph y=f(x)+cy = |f(x) + c| for a constant cc?

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