Cubic function graph sketching
When we sketch any graph we find out what its distinguishing features are and by plotting these
distinguishing features we are able to draw our graph accurately enough.
So what are the distinguishing features of a cubic function?
The graph of a hypothetical cubic function 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 is shown below:
𝑦
𝑦 − intercept
Point of inflection
𝑥
𝑥 − intercepts
Turning points / Stationary points
The distinguishing features are:
The 𝒙 − and 𝒚 − intercepts
Where the graph cuts the 𝑥 and 𝑦 axes respectively
These are found by setting 𝑦 = 0 and 𝑥 = 0 respectively.
The stationary points
Where the graph has a horizontal tangent.
An example of a stationary point that we are familiar with is that of a turning point.
But not all stationary points are turning points.
The stationary points are found by finding the points where the derivative of the curve is 0.
𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑝𝑜𝑖𝑛𝑡𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑤ℎ𝑒𝑟𝑒 𝑦 ′ = 𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 = 0.
The point of inflection
This is the point where the curve goes from being concave up to concave down or vice versa.
This is also the point where 𝑦 ′′ = 0.
A graph is A graph is
CONCAVE UP CONCAVE DOWN
when all of its when all of its
tangent lines are tangent lines are
below the curve. above the curve.
(Arms up) (Arms down)
Shape
The final distinguishing feature is the shape of the graph, subtle changes in the equation of
the function can make drastic changes to the shape of the curve.
1
, The graph on the left:
increases first then decreases and then increases again,
has two distinct turning points (stationary points)
The graph on the right:
initially decreases, then appears to flatten out, but then decreases again without ever
increasing
has no distinct turning point, but does have a point (3;27) where the tangent to the curve is
horizontal (definition of a stationary point)
Despite the differences, they both only have one 𝑥 − intercept, not the 3 that we saw in our
hypothetical example.
STEPS TO SKETCHING A CUBIC FUCTION:
STEP 1: Determine the BASIC SHAPE of the graph:
If 𝑎 > 0 then the graph will start off increasing from left to right
If 𝑎 < 0 then the graph will decrease initially
𝑎>0 𝑎<0
Why does the graph look the way it looks?
For 𝑎 > 0:
When 𝑥 is a negative number of a large magnitude, then 𝑦 will also be a high magnitude negative
number, on the other hand as 𝑥 gets very large 𝑦 will get very large too. This causes the tails on the
left and right to go off towards negative infinity and positive infinity, respectively.
This is because when we cube a large number it will dwarf the same number squared and cubing
preserves signs whereas squaring a number is always positive. Cubing a smaller number is closer in
value to squaring the same value.
Cubing a fraction on the other hand will have a lower magnitude than squaring the same fraction.
2
When we sketch any graph we find out what its distinguishing features are and by plotting these
distinguishing features we are able to draw our graph accurately enough.
So what are the distinguishing features of a cubic function?
The graph of a hypothetical cubic function 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑 is shown below:
𝑦
𝑦 − intercept
Point of inflection
𝑥
𝑥 − intercepts
Turning points / Stationary points
The distinguishing features are:
The 𝒙 − and 𝒚 − intercepts
Where the graph cuts the 𝑥 and 𝑦 axes respectively
These are found by setting 𝑦 = 0 and 𝑥 = 0 respectively.
The stationary points
Where the graph has a horizontal tangent.
An example of a stationary point that we are familiar with is that of a turning point.
But not all stationary points are turning points.
The stationary points are found by finding the points where the derivative of the curve is 0.
𝑆𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 𝑝𝑜𝑖𝑛𝑡𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑤ℎ𝑒𝑟𝑒 𝑦 ′ = 𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 = 0.
The point of inflection
This is the point where the curve goes from being concave up to concave down or vice versa.
This is also the point where 𝑦 ′′ = 0.
A graph is A graph is
CONCAVE UP CONCAVE DOWN
when all of its when all of its
tangent lines are tangent lines are
below the curve. above the curve.
(Arms up) (Arms down)
Shape
The final distinguishing feature is the shape of the graph, subtle changes in the equation of
the function can make drastic changes to the shape of the curve.
1
, The graph on the left:
increases first then decreases and then increases again,
has two distinct turning points (stationary points)
The graph on the right:
initially decreases, then appears to flatten out, but then decreases again without ever
increasing
has no distinct turning point, but does have a point (3;27) where the tangent to the curve is
horizontal (definition of a stationary point)
Despite the differences, they both only have one 𝑥 − intercept, not the 3 that we saw in our
hypothetical example.
STEPS TO SKETCHING A CUBIC FUCTION:
STEP 1: Determine the BASIC SHAPE of the graph:
If 𝑎 > 0 then the graph will start off increasing from left to right
If 𝑎 < 0 then the graph will decrease initially
𝑎>0 𝑎<0
Why does the graph look the way it looks?
For 𝑎 > 0:
When 𝑥 is a negative number of a large magnitude, then 𝑦 will also be a high magnitude negative
number, on the other hand as 𝑥 gets very large 𝑦 will get very large too. This causes the tails on the
left and right to go off towards negative infinity and positive infinity, respectively.
This is because when we cube a large number it will dwarf the same number squared and cubing
preserves signs whereas squaring a number is always positive. Cubing a smaller number is closer in
value to squaring the same value.
Cubing a fraction on the other hand will have a lower magnitude than squaring the same fraction.
2
