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INVESTIGATION: Relating gradient and original functions

Lesson

In Properties of graphs we saw many connections between the graph of a function and its derivative. This investigation has 3 puzzle activities to practice and affirm these connections.

 

Summary
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

 

Puzzle 1

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.

Puzzle 2

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.

Puzzle 3

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

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

Outcomes

2.3.18

construct and interpret position-time graphs, with velocity as the slope of the tangent

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