Equation of tangent line

Question

Find equation of the tangent line

f(x) = -4x - 4

at x=7

Answer 1

My guess would be there must be a typo in the question. It makes no sense as written. You can't have a tangent line to a line. They're the same line. There is no point of tangency.

Odds are you meant to write f(x) = -4x^2 -4 and the exponent 2 got left out.

then you want the tangent line where x=7 or at the point (7, -4(7^2)-4) = (7, -200)

the slope of the tangent line is the derivative: f'(x) = -8x = -8(7) = -56 = m

the tangent line is y=mx +b

y=-56x + b. solve for b by plugging in the point (7,-200)

-200=-56(7) + b

b = -200 + 392 = 192

y=-56x + 192 is the tangent line

Answer 2

To find an equation of the tangent line start with the point - slope form:

y - y₁ = m (x - x₁)

given the point (7, -32), so we will plug those values in for x₁ and y₁ in the above equation.

To find the slope, m, of the tangent line, find the derivative of the given function

f'(x) = -4x -4

f'(x) = -4, so m = -4.

Now, write the equation in point-slope form:

y - (-32)= -4 (x -7)

y+32=-4x+28

y=-4x+28-32

y=-4x-4