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

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

3. Features of the Slope Function Graph
Reading the features of the graph of the slope function from the original function -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
Turning Point
of the slope
function: where slopes turn from decreasing to increasing = min
slope function = 0 (cuts x-axis)
dy/dx= 0
Slope values
are increasing
→slope function
increasing
Slope values
are decreasing
→slope function
decreasing
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. The slope function is… f’(x) is… dy/dx is…-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
Slope values
are decreasing
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0
ORIGINAL FUNCTION
y = f(x) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
Slope values
are decreasing
are increasing
SLOPE FUNCTION
y = f’(x) x
dy/dx= 0; slope function = 0
dy/dx= 0; slope function = 0
Turning Point:
Decreasing to
increasing
= min pt

5. + + + + + + + 0 - - - - - - - - - - 0 + + + + + + +
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 x y
Slopes are negative
Slopes are positive
Slopes are positive -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
Slope Function above x-axis
Slope Function above x-axis
Slope Function below x-axis

6. At what rate is the slope function changing? f’’(x) is… d2y/dx2 is...-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
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. 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 x y
Slope values
are decreasing
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0
ORIGNAL FUNCTION
y = f(x) -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x y
Slope values
are decreasing
are increasing
Turning Point:
Decreasing to
increasing
= min pt
dy/dx= 0; slope function = 0
SLOPE FUNCTION
y = f’(x)

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

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

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