# Given that point (x, y) is on the graph of y = 4 - x², express the distance from (3, 4) to (x, y) as a function of x.

Given that point (x, y) is on the graph of y = 4 - x², express the distance from (3, 4) to (x, y) as a function of x.
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Viktor Wiley

Use the Distance Formula:
$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$
Let $\left({x}_{1},{y}_{1}\right)=\left(3,4\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left({x}_{2},{y}_{2}\right)=\left(x,y\right)$ so that:
$d={\sqrt{x-3}}^{2}+{\left(y-4\right)}^{2}\right\}$
Substitute $y=4-{x}^{2}$ (since (x,y) is on this graph):
$d=\sqrt{{\left(x-3\right)}^{2}+{\left(4-{x}^{2}-4\right)}^{2}}$
$d=\sqrt{{\left(x-3\right)}^{2}+{\left(-{x}^{2}\right)}^{2}}$
$d=\sqrt{{x}^{2}-6x+9+{x}^{4}}$
or
$d\left(x\right)=\sqrt{{x}^{4}+{x}^{2}-6x+9}$