Worked examples
The question
Question: find the end behavior of f(x) = x³ − 4x + 1.
f(x) = x3 − 4x + 1
1 Find the leading term
The leading term is the part with the biggest power of x.
It decides where the graph goes at the far ends.
leading term: the biggest power of x
The biggest power of x here is x³.
So the leading term is x³.
leading term : 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.
power 3: odd → ends go opposite ways
plus in front → right end goes up
Careful!
An odd power keeps the sign of x.
A big negative x gives the opposite of a big positive x.
Careful: odd power: one end up, one end down
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.
= right end: x → ∞, f(x) → ∞
Far to the left, x is a big negative number.
x³ goes down: f(x) goes to minus infinity.
= left end: x → −∞, f(x) → −∞
We can see it on a small sketch of x³.
Left end down, right end up.
3 Check
Let's check with a big x: x = 10 in x³ gives 1000: big and positive, so up.
x = 10: 103 = 1000 up
And x = −10: −1000, a big negative number: down.
It matches!
x = −10: (−10)3 = −1000 down
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.
Answer = left end: x → −∞, f(x) → −∞
▶ Watch it solved
The question
Question: find the end behavior of f(x) = 4 + x − 2x³.
f(x) = 4 + x − 2x3
1 Find the leading term
The leading term is the part with the biggest power of x.
It decides where the graph goes at the far ends.
leading term: the biggest power of x
Careful!
The leading term is not just the first part written.
It is the biggest power: −2x³.
Careful: not leading term 4 ✗
The biggest power of x here is x³.
So the leading term is −2x³.
leading term : −2x3
2 Read the ends
The power 3 is odd.
So the two ends go opposite ways.
The number in front is negative, so the right end goes down.
power 3: odd → ends go opposite ways
minus in front → right end goes down
Far to the right, x gets bigger and bigger.
−2x³ goes down for ever.
Books write: as x goes to infinity, f(x) goes to minus infinity.
= right end: x → ∞, f(x) → −∞
Far to the left, x is a big negative number.
−2x³ goes up: f(x) goes to infinity.
= left end: x → −∞, f(x) → ∞
We can see it on a small sketch of −2x³.
Left end up, right end down.
3 Check
Let's check with a big x: x = 10 in −2x³ gives −2000: big and negative, so down.
x = 10: −2 × 103 = −2000 down
And x = −10: 2000, a big positive number: up.
It matches!
x = −10: −2 × (−10)3 = 2000 up
Let's look back.
First: find the leading term.
Then: read the ends.
Last: check.
Answer = left end: x → −∞, f(x) → ∞
▶ Watch it solved
The question
Question: graph f(x) = x²(x − 3).
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.
leading term : x2 × x = 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.
power 3: odd → ends go opposite ways
plus in front → 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.
= right end: x → ∞, f(x) → ∞
Far to the left, x is a big negative number.
x³ goes down: f(x) goes to minus infinity.
= left end: x → −∞, f(x) → −∞
3 Find the zeros
x² is 0 when x = 0.
Its power is 2: even, so the graph touches.
x2 → x = 0, power 2: touches
f(x) is 0 when one of the brackets is 0.
Two numbers multiply to 0 only if one of them is 0.
A × B = 0 → A = 0 or B = 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.
odd power: crosses the x-axis
even power: touches it, turns back
(x − 3) is 0 when x = 3.
Its power is 1: odd, so the graph crosses.
(x − 3) → x = 3, power 1: crosses
Careful!
At each zero, look at the power of its bracket.
Odd crosses the axis.
Even only touches it.
Careful: odd power: crosses, even power: touches
4 Find the y-intercept
The graph crosses the y-axis where x is 0.
Put 0 in place of every x.
f(0) = (0)2(0 − 3)
So the y-intercept is (0, 0).
A bracket is just x, so it is 0.
f(0) = 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.
f(1) = 12(1 − 3) = 1 × (−2) = −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.
▶ Watch it solved
