To find the equation of the tangent line to the graph of a function at a specific point, we can use the point-slope formula of a line. The point-slope form of a line can be written as follows:
y - y₁ = m(x - x₁)
where:
- (x₁, y₁) is point on the line
- m is the slope of the line
In this case, we are given the point (6,-2) (since the tangent touches the graph at x=6 where g(6) = -2) and the slope of the tangent line at this point is 5 (since the derivative of the function g(x) at x=6, g'(6), is 5).
Therefore,
x₁ = 6,
y₁ = -2, and
m = 5.
Substitute these values into the equation of the line to get:
y - (-2) = 5(x - 6)
Simplify the above equation to get:
y + 2 = 5x - 30
Subtracting by 2 from both sides of the equation to get:
y = 5x - 30 - 2
Further simplify to get:
y = 5x - 32
So the equation of the tangent line to the graph at x = 6 is y = 5x - 32.