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The function is defined as .What is the -coordinate of the point on the function's graph that is closest to the origin?
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Q. The function is defined as .What is the -coordinate of the point on the function's graph that is closest to the origin?
- Define Distance Formula: To find the point on the graph of the function that is closest to the origin, we need to minimize the distance from any point on the graph to the origin . The distance between the point and the origin can be found using the distance formula: .
- Substitute : Substitute with into the distance formula: .
- Expand Expression: Expand the expression inside the square root: .
- Combine Like Terms: Combine like terms inside the square root: .
- Minimize Expression: To find the minimum distance, we can minimize the expression inside the square root since the square root function is increasing. This is equivalent to minimizing the quadratic expression . To find the minimum of a quadratic function, we can use the vertex formula , where the quadratic is in the form .
- Apply Vertex Formula: Apply the vertex formula to the quadratic expression . Here, and . So, .
- Find x-coordinate: The -coordinate of the point on the graph of the function that is closest to the origin is . This is the final answer.
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