1. Relating Graphs of f and f’

3. Changing from concavity from up to down Changing from concavity from down to up At a relative minimum (cusp) and concave down At a relative maximum (cusp) and concave up At a relative minimum (smooth curve) and concave up At a relative maximum (smooth curve) and concave down Decreasing (has a negative slope) and concave down Decreasing (has a negative slope) and concave up Increasing (has a positive slope) and is concave down Increasing (has a positive slope) and concave up And the second derivative is… Then the derivative is… (and the graph of the derivative …) When the function is…(and the graph of the function…)

4. Zero – crossing x-axis from above to below axis At a maximum, could be positive or negative Changing from concavity from up to down Zero -Crossing x-axis from below to above At a minimum, could be positive or negative Changing from concavity from down to up Negative/undefined Undefined At a relative minimum (cusp) and concave down Positive/undefined Undefined At a relative maximum (cusp) and concave up Positive Zero (crossing the x-axis from negative to positive) At a relative minimum (smooth curve) and concave up Negative Zero (crossing the x-axis from above to below) At a relative maximum (smooth curve) and concave down Negative Negative (the graph of f’ is below the x-axis and has a negative slope) Decreasing (has a negative slope) and concave down Positive Negative (the graph of f’ is below the x-axis and has a positive slope) Decreasing (has a negative slope) and concave up Negative Positive (the graph of f’ is above the x-axis and has a negative slope) Increasing (has a positive slope) and is concave down Positive Positive (the graph of f’ is above the x-axis and has a positive slope) Increasing (has a positive slope) and concave up And the second derivative is… Then the derivative is… (and the graph of the derivative …) When the function is…(and the graph of the function…)