Question

find an equation of the tangent line to the graph of ( y = g(x) ) at ( x = 6 ) if ( g(6)=-4 ) and ( g^{prime}(6)=2 ). (enter your answer as an equation in terms of ( y ) and ( x ).)

Explanation:

Step1: Recall the point - slope form of a line

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line.

Step2: Identify the values of \(x_1,y_1\) and \(m\)

We are given that \(x = 6\), \(g(6)=-4\), so \(x_1 = 6\) and \(y_1=-4\). Also, \(g^{\prime}(6) = 2\), and the slope \(m\) of the tangent line at \(x = 6\) is \(m = g^{\prime}(6)=2\).

Step3: Substitute the values into the point - slope form

Substitute \(x_1 = 6\), \(y_1=-4\) and \(m = 2\) into \(y - y_1=m(x - x_1)\).
We get \(y-(-4)=2(x - 6)\).

Step4: Simplify the equation

\(y + 4=2x-12\).
Then \(y=2x-16\).

Answer:

\(y = 2x-16\)