SECT 3-8B RELATING GRAPHS

Handout: Relating Graphs

Relating GraphsIf the function is increasing, then the graph of the derivative is above the x-axis

If the function is decreasing, then the graph of the derivative is below the x-axis

If the function has extrema, then the graph of the derivative is crossing the x-axis

If the function changes concavity, then the graph of the derivative has an extrema

Relating GraphsIf the graph of the derivative is above the x-axis (positive) then the function is increasing

If the graph of the derivative is below the x-axis (negative) then the function is decreasing

If the graph of the derivative has an zero then the function has a possible max or min

Relating GraphsIf the graph of the derivative is decreasing (negative slope) then the function is concave down

If the graph of the derivative is increasing (positive slope) then the function is concave up

If the graph of the derivative has an extrema then the function has a possible point of inflection

Using Number lines•Antiderivatives: f ′ graph → f graph •1. Make an f ′ number line by using the location or position of the points on the f ′ graph. This does not involve the slopes of f ′.•2. Make an f ′′ number line by using the slopes of the f ′ graph.•3. Combine information from both number lines to graph f . If no starting point is given, you are free to shift the graph vertically. 

7)

The graph of is given above. The domain of f is a) For what value(s) of x, is the graph of f increasing? Justify your answer.

b) For what value(s) of x, does the graph of f have a relative maximum? Justify your answer

33 x)(' xfy 3,3

33 x

7)

The graph of is given above. The domain of f is c) For what value(s) of x, is the graph of f concave up and concave down? Justify your answer.

d) For what value(s) of x, does the graph of f have inflection points? Justify your answer.

33 x)(' xfy 3,3

33 x

7) Sketch a possible graph of )(xf

8) The graph of f ′ is shown, sketch a possible graph of f with a starting point of (0,1)

x y

x y

9) Use the graph of f ′ shown to sketch a graph of f ′′ and a possible graph of f .f ′′ f ′ f

x

y

x

y

x

y

HomeworkWorksheet 3-8B