+0

# Graphing question

0
35
1

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

Jan 1, 2023

#1
+2541
+1

We want to minimize $$\sqrt{x^2+ y^2}$$

Substituting $$y = {1 \over 2}\times(x^2 - 8)$$, we get $$\sqrt{x^2 + {x^4 \over 4} - 4x^2 + 16}$$, which simplifies to $$\sqrt{{x^4 \over 4} - 3x^2 + 16}$$

Now, let $$z = x^2$$. We have $$\sqrt{{z^2 \over 4} - 3z + 16}$$. Because this is a quadratic, the minimum occurs at $$-{b \over 2a} = -{3 \over 2 \times {0.25}} = 6$$

Substituting this in gives us $$\sqrt{{6^2 \over 4} - 3\times 6^2 + 16} =\color{brown}\boxed{ \sqrt{7}}$$

Jan 1, 2023