Graphing a function when you have a graph of the derivative and possibly other information:
FIRST
Plot any value of the function (f(x))
SECOND Determine where the function is increasing and decreasing
If graph of derivative is positive (above the x-axis) then function is increasing
If graph of derivative is negative (below the x-axis) then function is decreasing
THIRD
Determine where you have local minimums, maximums and “S”s
Where the graph of the derivative equals zero, then
-if it goes from negative to positive, you have a minimum
-if it goes from positive to negative you have a maximum
-if it goes positive to positive, you get an “upward S”
-if it goes negative to negative you get a “downward S”
FOURTH If the graph of the derivative is horizontal, then the slope is constant and the graph of the
function is a straight line over that interval (which also means that it is neither concave up or
down)
FIFTH
Determine where the graph is concave up or down
--If the slope of the graph of the derivative (which is the second derivative) is positive (the
graph of the derivative is increasing), then the slope is positive and the function is
concave up
--If the slope of the graph of the derivative (which is the second derivative) is negative (the
graph of the derivative is decreasing), then the slope is negative and the function is
concave down
SIXTH
Draw the graph
INFLECTION POINTS: where the function goes from concave up to down, or down to up. Not
where it goes from up to up, or down to down, or if it is neither on either side (the derivateive
is a constant or the graph of the derivative is a straight line)
LOCAL MINs and MAXs—Identified in “THIRD “ above