Step by step
The question
Question: graph f(x) = x²(x − 3).
1 Find the leading term
Multiply the x's of the brackets.
Each bracket gives its x, to its power.
So the leading term is x³.
2 Read the ends
The power 3 is odd.
So the two ends go opposite ways.
The number in front is positive, so the right end goes up.
Far to the right, x gets bigger and bigger.
x³ goes up for ever.
Books write: as x goes to infinity, f(x) goes to infinity.
Far to the left, x is a big negative number.
x³ goes down: f(x) goes to minus infinity.
3 Find the zeros
x² is 0 when x = 0.
Its power is 2: even, so the graph touches.
f(x) is 0 when one of the brackets is 0.
Two numbers multiply to 0 only if one of them is 0.
The power of each bracket is called the multiplicity of that zero.
Odd: the graph crosses the x-axis there.
Even: it only touches the axis and turns back.
(x − 3) is 0 when x = 3.
Its power is 1: odd, so the graph crosses.
Careful!
At each zero, look at the power of its bracket.
Odd crosses the axis.
Even only touches it.
4 Find the y-intercept
The graph crosses the y-axis where x is 0.
Put 0 in place of every x.
So the y-intercept is (0, 0).
A bracket is just x, so it is 0.
5 Draw the graph
Draw the grid.
Plot the zeros on the x-axis.
Start at the left end, going down.
At 0 it touches and turns.
At 3 it crosses.
The right end goes up.
6 Check
Let's check one point: x = 1.
We get −2.
(1, −2) is below the axis, right on our curve.
It works!
Let's look back.
First: find the leading term.
Then: read the ends.
Then: find the zeros.
Then: find the y-intercept.
Then: draw the graph.
Last: check.