In this particular case, I am trying to find all points $(x,y)$ on the graph of $f(x)=x^2$ with tangent lines passing through the point $(3,8)$.
Now then, I know the graph of $x^2$. What now?
[Math] Finding all points on a graph with tangent lines passing through a particular point
algebra-precalculuscalculusgraphing-functions
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Best Answer
A point on the graph is $(x,x^2)$. The slope from that point to $(3,8)$ is given by: $\frac{x^2-8}{x-3}$. This has to be equal to the derivative at the point for it to be a tangent. So: $$\frac{x^2-8}{x-3}=2x$$ $$x^2-8=2x^2-6x$$ $$0=x^2-6x+8$$ $$0=(x-2)(x-4)$$ So $x=2$ or $x=4$. So the points are $(2,4)$ and $(4,16)$.