A tangent is commonly written in the form y=mx+b

Generally to find the equation of the tangent of a function at a given point you find the derivative, or the slope of the function at the given point, and then you substitute x and y values to solve for b. But by writing the equation differently you can skip the last step.

Instead this method states that once you have calculated the derivative of the function you approach the problem by solving for the horizontal and vertical shift.

Thus given f(x), the equation of the tangent to the function f(x) is:

y=m(x-x)+(f(x))

heres a quick example.

f(x)= x^2-6x+17

dy/dx = 2x-6 (used the power rule for calculating derivatives, this is "m" in y=mx+b)

thus the equation of the tangent:

y=(2x-6)(x-x)+(x^2 - 6x+17)

this solves for the equations of all tangents to the function f(x).

substituting a value for x into (2x-6) solves for the slope at that given x value.

Substituting the same value for x into the second x only * of (x-x) will apply a horizontal shift. The first x remains variable.

and substituting x into (x^2 - 6x+17) will apply the vertical shift.

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