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# Graphing

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What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 8)?

Feb 6, 2023

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By the distance formula, the distance from the origin to the graph is $$\sqrt{(y-0)^2 + (x-0)^2} = \sqrt{y^2 + x^2}$$

We know that $$y = {1 \over 2}x^2 - 4$$, meaning $${1 \over 2}x^2 = y + 4$$, so $$x^2 = 2y + 8$$

Substituting this gives us $$\sqrt{y^2 + 2y + 8}$$

The minimum value occurs at $$y = -{b \over 2a}$$, which is -1.

This means the minimum is $$\sqrt{(-1)^2 + 2(-1) + 8} = \color{brown}\boxed{\sqrt{7}}$$

Feb 6, 2023