When using calculus to explore the properties of graphs, there were many connections made between the graph of a function and its derivative. This investigation has three puzzle activities to practice identifying 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 pieces, shuffle and then try to complete the puzzle by making 12 pairs. This puzzle is also a great way to select partners for puzzle 2. Hand out the puzzle pieces and silently try to find a match amongst your classmates.
The second puzzle has a set of 12 functions graphs, 12 derivative graphs and descriptions. Cut out the puzzle pieces, shuffle and then try to complete the puzzle by making 12 sets of a function, its derivative and the description for both the function and derivative. This puzzle is a good activity to complete in pairs and 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, shuffle and then try 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
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
