Step by step
The question
Question: sketch the graph of y = x³ − 6x² + 9x.
1 Where it crosses the axes
Where it crosses the y-axis: put x = 0.
We get y = 0.
Where it crosses the x-axis: y = 0.
Every term has an x: take it out.
Two numbers that multiply to 9 and add to −6: −3 and −3, the same number twice.
So x = 0, or 3.
2 Turning points
The slope: f′(x) = 3x² − 12x + 9, by the power rule.
Turning points: where the slope is 0.
Take out the 3.
We need two numbers that multiply to 3 and add to −4: −1 and −3.
A product is 0 only when one of its parts is 0.
So x = 1 or x = 3.
To tell max from min: f″(x) = 6x − 12.
f″(1) is negative: a cap, a maximum.
Its y is 4.
f″(3) is positive: a cup, a minimum.
Its y is 0.
So: a maximum at (1, 4), and a minimum at (3, 0).
3 Inflection points
The bending changes where f″ is 0.
So x = 2.
Its y is 2.
Careful!
Turning points are where f′ is 0.
The inflection point is where f″ is 0.
They are different points.
4 The two ends
Far out, the x³ term wins.
On the right the graph goes up for ever.
On the left it goes down.
5 Draw the graph
Now plot the key points.
Where it crosses the x-axis.
The maximum.
The minimum.
The inflection point.
Join them with a smooth curve: up to the max, down to the min, then up for ever.
6 Check
Let's check one more point: x = 4 gives y = 4.
It sits on our curve.
Right!
Let's look back.
First: where it crosses the axes.
Then: turning points.
Then: inflection points.
Then: the two ends.
Then: draw the graph.
Last: check.