Find an equation for the tangent line to the graph

Question

Find an equation for the tangent line to the graph of y = 3x² − 7 at x = −1.

Answer 1

The equation of the tangent line to a function at a given point can be found using the point-slope form of a line, which is given by:

y - y_1 = m(x - x_1)

where ((x_1, y_1)) are the coordinates of the point on the function and (m) is the slope of the tangent line at that point.

The slope of the tangent line is the derivative of the function evaluated at the given point. The derivative of the function (y = 3x^2 - 7) is (y’ = 6x). Evaluating this at (x = -1) gives (m = 6(-1) = -6).

The y-coordinate of the point on the function where (x = -1) is (y = 3(-1)^2 - 7 = -4).

Substituting these values into the point-slope form gives the equation of the tangent line:

y - (-4) = -6(x - (-1))

Simplifying this gives:

y = -6x - 10

So, the equation of the tangent line to the graph of (y = 3x^2 - 7) at (x = -1) is (y = -6x - 10)

Answer 2

y = 3x^2 -7 has slope = y' = 6x

at x=-1, the slope = 6(-1) = -6

at x=-1, y = 3(1)-7 = -4 or the point (-1,-4)

a tangent line would have slope =-6 = (y+4)/(x+1)

multiply by x+1 to get

y+4 = -6x -6

subtract 4 from both sides to get

y = -6x -10

is the equation of the tangent line when x=-1, in slope intercept form