The document discusses features of functions with the form y=x^3 including: - The original function f(x) is cubic and increases then decreases. - Its derivative f'(x) represents the slope and is U-shaped, cutting the x-axis at points of inflection. - The second derivative f''(x) represents the rate of change of slope and directly relates to the original function's concavity.

1. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Increasing
Increasing
Decreasing
Features of +x3
Graphs
1
Stationary
Stationary
The original function is… f(x) is… y
is…

2. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
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1
2
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x
y
Investigate the tangents of +x3
Graphs
2
The slope function is… f’(x) is… dy/dx is…
Slope values
are decreasing
Slope values
are increasing
Point of
Inflection =
slopes stop
decreasing
and start
increasing

3. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are decreasing
→slope function
decreasing
Features of the Slope Function Graph
3
Reading the features of the graph of the slope function from the
original function
Slope values
are increasing
→slope function
increasing
Turning
Point
of the slope
function:
where
slopes turn
from
decreasing
to increasing
= min
slope function = 0 (cuts x-axis)
dy/dx= 0
dy/dx= 0
slope function = 0 (cuts x-axis)
Slope Function: U shaped (positive cubic graph will have positive derivative graph)
Minimum point at same x value as the point of inflection
Cuts x-axis at the x values of the turning points

4. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are decreasing
Slope values
are increasing
4
The slope function is… f’(x) is… dy/dx is…
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are decreasing
Slope values
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0
dy/dx= 0; slope function = 0
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0
SLOPE FUNCTION
y = f’(x)
ORIGINAL FUNCTION
y = f(x)
x

5. 5
Also, we can read where the slope function is above and below the x-
axis from the original function
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slopes are
positive
Slope
Function
above x-axis
Slopes are
negative
Slope
Function
below x-axis
Slopes are
positive
Slope
Function
above x-axis
+ + + + + + + 0 - - - - - - - - - - 0 + + + + + + +

6. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
6
At what rate is the slope function changing? f’’(x) is… d2y/dx2 is...
How fast is
the rate of
decrease of
the slopes?
How fast is
the rate of
increase of
the slopes?
Finding the rate of change of the rate of change…. Finding the second derivative

7. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are decreasing
Slope values
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0
7
A step further to investigate the tangents of the slope function.
Second Derivative Function is… f’’(x) is… d2y/dx2 is...
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are decreasing
Slope values
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0
dy/dx= 0; slope function = 0
ORIGNAL FUNCTION
y = f(x)
SLOPE FUNCTION
y = f’(x)

8. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are decreasing
Slope values
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0
SLOPE FUNCTION
y = f’(x)

9. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
y
Slope values
are increasing
→Second Derivative
Function is increasing
Slope values
are increasing
→Second Derivative
Function is increasing
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
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1
2
3
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5
6
x
y
SECOND DERIVATIVE
FUNCTION
y = f’’(x)
Slope=0 (d2y/dx2 = 0)
Second Derivative Function =0
(cuts x-axis)
SLOPE FUNCTION
y = f’(x)

10. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
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x
y
10
Original Function, First Derivative Function, Second Derivative Function
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
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1
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x
y
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
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1
2
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x
y
y = f’’(x)
𝒍𝒐𝒄𝒂𝒍 𝒎𝒂𝒙
=
𝒅𝟐
𝒚
𝒅𝒙𝟐
< 𝟎
𝒍𝒐𝒄𝒂𝒍 𝒎𝒊𝒏
=
𝒅𝟐
𝒚
𝒅𝒙𝟐
> 𝟎
y = f’(x)
y = f(x)
𝑻𝒖𝒓𝒏𝒊𝒏𝒈 𝑷𝒐𝒊𝒏𝒕𝒔 𝒐𝒇
𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒂𝒕
𝒅𝒚
𝒅𝒙
= 𝟎