A. Find the slope of the tangent line to the graph of f at...

a. Find the slope of the tangent line to the graph of $f$ at the given point.
b. Find the slope-intercept equation of the tangent line to the graph of $f$ at the given point.
$$f(x)=4 x^{2} \text { at }(-2,16)$$

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation, written as y = mx + b, expresses the line in terms of its slope (m) and y-intercept (b). This form is particularly useful for quickly identifying the rate of change and where the line crosses the y-axis, making it a standard representation in linear equations and tangent line problems.

Power Rule

The power rule is a basic differentiation technique used to find the derivative of functions of the form x^n. It states that the derivative of x^n is n*x^(n-1), making it a quick and effective method for computing derivatives of polynomial functions, which is essential in finding slopes of tangent lines.

Tangent Line

The tangent line to a function at a specific point is the straight line that just 'touches' the curve at that point and has the same slope as the function at that point. It serves as the best linear approximation of the function near that point and is crucial for understanding the local behavior of the function.

Derivative

The derivative of a function represents its instantaneous rate of change and is the fundamental tool used to determine the slope of the tangent line to a curve at any given point. It is computed using differentiation techniques and provides the slope of the curve at specific points, serving as the foundation for concepts such as instantaneous velocity or optimization in various fields.

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