A2.2

generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, $f(x)$, for various values of $x$ (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, $f^{\prime }(x)$ or $\frac{dy}{dx}$, and make connections between the graphs of $f(x)$ and $f^{\prime }(x)$ or $y$ and $\frac{dy}{dx}$ [e.g., when $f(x)$ is linear, $f^{\prime }(x)$ is constant; when $f(x)$ is quadratic, $f^{\prime }(x)$ is linear; when $f(x)$ is cubic, $f^{\prime }(x)$ is quadratic] Sample problem: Investigate, using patterning strategies and graphing technology, relationships between the equation of a polynomial function of degree no higher than 3 and the equation of its derivative.