Math Problem Statement
Are you sure you look at the grahs and choose the right one?
Solution
To identify which diagram correctly represents the parametric curve given by:
we can analyze the behavior of and over the interval to understand the path of the curve.
-
When :
- The curve starts at the point .
-
When :
- The curve ends at the point .
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Intermediate values of :
- For between and , varies from to (positive values) and then back to . Therefore, will range from to and back to .
- varies from to .
From this information, we can deduce that the curve traces a semicircle, moving from to , with values remaining negative or zero. This matches a left-facing semicircle centered at the origin.
Diagram Choice:
- The correct option is (B), which shows a left-facing semicircle from to .
Let me know if you'd like more details on any step.
Additional Questions:
- How can you determine the direction of a parametric curve?
- Why does the value of only go up to here?
- What would happen if the range of extended beyond ?
- Can parametric curves represent other shapes besides circles?
- How would the graph change if instead of ?
Tip: When analyzing parametric curves, checking endpoint values and the general shape by looking at intermediate values can help quickly identify the curve’s form.
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Trigonometric Functions
Curve Sketching
Formulas
x = -sin(t)
y = cos(t)
0 ≤ t ≤ π
Theorems
Behavior of Sine and Cosine Functions over an Interval
Suitable Grade Level
Grades 10-12
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