13. [-/1 Points] DETAILS MY NOTES SCALCET9 2.7.028. Find an equation of the tangent line to the graph of \( y=g(x) \) at \( x=2 \) if \( g(2)=-3 \) and \( g^{\prime}(2)=5 \). (Enter your answer as an equation in terms of \( y \) and \( x \).) ASK YOUR TEACHER

To find the equation of the tangent line to the graph of \( y = g(x) \) at the point where \( x = 2 \), we start with the point-slope form of the line equation, which is \( y - y_1 = m(x - x_1) \). Here, \( m \) is the slope of the tangent line, which is \( g'(2) = 5 \), and the point through which the line passes is \( (x_1, y_1) = (2, -3) \). Plugging in these values, we get: \[ y - (-3) = 5(x - 2) \] This simplifies to: \[ y + 3 = 5x - 10 \] \[ y = 5x - 13 \] So, the equation of the tangent line is \( y = 5x - 13 \). In the world of calculus, the concept of tangent lines is crucial as it gives insights into how functions behave near specific points. This is particularly useful in fields like physics and engineering, where you might need to devise instant velocities or rates of change. Knowing how to find and apply the equation of a tangent line can unlock a treasure chest of practical applications in real-world scenarios like optimizing trajectories or analyzing motion. As you dive deeper into mathematics, avoid the common pitfall of confusing the tangent line’s slope with the function's average rate of change over an interval. A tangent line reflects instantaneous behavior at a specific point, while the average rate of change covers an interval. Always remember: the derivative gives us the slope of the tangent, not the average slope!