Find an equation of the tangent line to the graph of #y=g(x)# at #x=5# if #g(5) = -3# and #g'(5) = 4#? Can you explain this to me in detail?

Question

Samantha Adkison · Accepted Answer

#y = 4x - 23#

This tells us that when #x = 5#, #y = -3#. Also, when the derivative equals #5#, the slope of the tangent is #4#.

This is all we need to find the equation of the tangent.

#y - y_1= m(x- x_1)#

#y- (-3) = 4(x - 5)#

#y+ 3 = 4x - 20#

#y = 4x - 23#

Hopefully this helps!

Emilia Aguilar · Answer

undefined To find the equation of the tangent line to the graph of y=g(x) at x=5, we need to use the point-slope form of a linear equation.

The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.

Given that g(5) = -3, we know that the point (5, -3) lies on the graph of g(x).

Also, given that g'(5) = 4, we know that the slope of the tangent line at x=5 is 4.

Using the point-slope form, we can substitute the values into the equation:

y - (-3) = 4(x - 5)

Simplifying, we get:

y + 3 = 4x - 20

Rearranging the equation to the standard form, we have:

4x - y = 23

Therefore, the equation of the tangent line to the graph of y=g(x) at x=5 is 4x - y = 23.