Let 𝑓(9)=−5 and 𝑓′(9)=8. Then the equation of the tangent line to the graph of 𝑦=𝑓(𝑥) at 𝑥=9 is 𝑦= ?

Answer 1

Given f(9) = -5 on the graph of the function y = f(x).

Also given that f '(9) = 8 which means that the derivative of the function at the point x = 9 is 8.

What this really means is that the slope (m) of the tangent line to the function y = f(x) at the point (9, -5) is 8.

To find the equation of that tangent line let's use y = mx + b (the slope-intercept form for the equation of any non-vertical line.

So replacing m with 8 we get y = 8x + b as the equation of the tangent line we need. All we need to do now is find b, the y-intercept.

So substituting 9 for x, -5 for y, and 8 for m we get,

-5 = 8(9) + b. Now solve for b

-5 = 72 + b and subtracting 72 from both sides,

b = -77

So the equation of the tangent line is y = 8x - 77

Answer 2

y-f(9) = f'(9)(x-9) is the point slope form of the tangent line.