There are many important connections between the graph of a function and its derivative, some of these connections have been explored in previous lessons. This investigation has 4 puzzle activities to practice and affirm these connections.
| Feature in function | Property of derivative |
|---|---|
| Function of degree n | Derivative is of degree n-1 |
| Increasing | f'(x)>0, the derivative graph is above the x-axis |
| Decreasing | f'(x)<0, the derivative graph is below the x-axis |
| Maximum turning point | f'(x)=0, the derivative graph has an x-intercept and crosses from above the x-axis to below |
| Minimum turning point | f'(x)=0, the derivative graph has an x-intercept and crosses from below the x-axis to above |
| Stationary point of inflection | f'(x)=0, the derivative graph has an x-intercept and just touches the x-axis forming a local maximum or minimum |
| Concave - slope decreasing | f'(x) has a negative slope |
| Convex - slope increasing | f'(x) has a positive slope |
| General point of inflection - change in concavity | f'(x) has a local maximum or minimum |
The first puzzle has a set of 24 graphs. There are 12 functions to be paired with their derivative. Cut out the puzzle piece and work to complete the puzzle by making 12 pairs. This puzzle can serve as a class activity to select partners for puzzle 2. Hand out a puzzle piece to each student and silently look for a match amongst your classmates.
The second puzzle has a set of 12 function graphs, 12 derivative graphs and 24 descriptions of each type of graph. Cut out the puzzle pieces and work to complete the puzzle by making 12 sets of a function, its derivative and the description for both graphs. This puzzle can be completed in pairs to discuss findings or as a solitary revision activity.
The third puzzle is a set of triangular puzzle pieces that when assembled form a large hexagon. Cut out the puzzle pieces and work to complete the puzzle by matching the function and its derivative along the edges of the triangles.
The fourth puzzle, just like puzzle 3, is a set of triangular puzzle pieces that when assembled form a large hexagon. This time the puzzle involves exponential and trigonometric functions and their derivatives. Cut out the puzzle pieces and work to complete the puzzle by matching the function and its derivative along the edges of the triangles.
Puzzle 1 pieces
Puzzle 2 pieces
Puzzle 3 pieces
Puzzle 4 pieces
Answers to puzzles can be found here.
Solution puzzle 1
Solution puzzle 2
| Function graph | Derivative graph | Function description | Derivative description | Equation |
|---|---|---|---|---|
| F1 | D10 | f7 | d12 | y=(x-1)(x-7) |
| F2 | D8 | f2 | d4 | y=0.5(x-2)^3-3 |
| F3 | D11 | f11 | d7 | y=0.8\left(0.75\right)^x |
| F4 | D9 | f5 | d10 | y=0.4x^4-1.6x^3+3 |
| F5 | D7 | f4 | d9 | y=4x+3 |
| F6 | D1 | f8 | d11 | y=0.25\left(\frac{x^3}{3}-x^2-8x+8\right) |
| F7 | D6 | f1 | d5 | y=\frac{-x^3}{3}+3x^2-5x-5 |
| F8 | D2 | f6 | d3 | y=\frac{x^5}{25}-\frac{3x^3}{5} |
| F9 | D12 | f10 | d6 | y=\frac{7}{0.4x^2+1} |
| F10 | D4 | f12 | d1 | y=4\left(1.2\right)^x |
| F11 | D5 | f3 | d8 | y=-\left(x-3\right)^2 |
| F12 | D3 | f9 | d2 | y=\frac{1}{16}\left(-3x^4+8x^3+30x^2-72x+16\right) |
Solution puzzle 3
Solution puzzle 4
