To solve this problem, we need to find the equation of the tangent line to the graph of $y = g(x)$ at the point $x = 2$.

Given:

• $g(2) = -4$
• $g'(2) = 3$

The equation of the tangent line at a point $(x_0, y_0)$ is given by:

$y - y_0 = m(x - x_0)$

Where $m$ is the slope of the tangent line, and $(x_0, y_0)$ is the point of tangency.

Here:

• $x_0 = 2$
• $y_0 = g(2) = -4$
• $m = g'(2) = 3$

Substitute these values into the equation:

$y - (-4) = 3(x - 2)$

Simplify this to get the equation in standard form:

$y + 4 = 3(x - 2)$

$y + 4 = 3x - 6$

$y = 3x - 10$

So, the equation of the tangent line is:

$y = 3x - 10$

Would you like any further details on this solution? Here are some related questions to deepen your understanding:

1. What does the derivative $g'(x)$ represent geometrically?
2. How do you find the slope of a tangent line if given a function's graph?
3. How would the equation change if $g'(2)$ was negative?
4. Can you derive the equation of the tangent line using a different method?
5. How does the tangent line equation relate to linear approximations?

Tip: Remember, the slope of the tangent line is always equal to the value of the derivative at that specific point.