[Math] Finding all tangent lines that pass through a specific point (not the origin)

calculusderivatives

I was given the function $y = x^3-x$ and told to find all tangent lines that pass through $(-2,2)$. Not sure what steps to take past finding the derivative.

Best Answer

We find the equation of the tangent line to $f(x) = x^3-x$ at the point $(k, k^3-k)$.

First, the derivative gives the slope $$f'(x) = 3x^2 -1$$

So we have a line with slope $3k^2-1$ that goes through the point $(k,k^3-k)$. What is its equation? We use the point-slope form

$$y-(k^3-k) = (3k^2-1)(x-k)$$

$$y-k^3 +k = 3k^2x-x-3k^3+k$$

$$y+2k^3 = 3k^2x-x$$

We want to know which values of $k$ make this line pass through $(x,y)=(-2,2)$, so we plug in those values:

$$2+2k^3=3k^2(-2)-(-2)$$

$$2k^3 = -6k^2$$

$$k^3 = -3k^2$$

$$k=0 \,\,\,\,\text{ or }\,\,\, k=-3$$

This means that there are two tangent lines that work, one at $(0,0)$, and the other at $(-3,-24)$. You should be able to find the equations of these lines easily if necessary.

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