Question

Find an equation of the tangent line to the graph of \( y=g(x) \) at \( x=6 \) if \( g(6)=-2 \) and \( g^{\prime}(6)=5 \). (Enter your answer as an equation in terms of \( y \) and \( x \).)

Answer 1

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.