Worked examples
The question
Question: here are three forms of one parabola: y = x² − 6x + 5, (x − 1)(x − 5), and (x − 3)² − 4.
What does each form show?
y = x2 − 6x + 5
= (x − 1)(x − 5)
= (x − 3)2 − 4
1 Standard form: the y axis
Standard form: x squared, then x, then a plain number.
y = x2 − 6x + 5
On the y axis, x is 0.
With x = 0, every x part is 0.
Only the plain number is left: 5.
on the y axis, x = 0: y = 0² − 6 × 0 + 5 = 5
So it crosses the y axis at (0, 5).
y axis : (0, 5)
2 Factored form: the x axis
Factored form: a product of brackets.
y = (x − 1)(x − 5)
On the x axis, y is 0.
So the product must be 0.
on the x axis, y = 0: (x − 1)(x − 5) = 0
A product is 0 only when one of its parts is 0.
x − 1 is 0 when x = 1.
x − 5 is 0 when x = 5.
A × B = 0 → A = 0 or B = 0
x − 1 = 0 → x = 1
x − 5 = 0 → x = 5
Careful!
Not −1.
x − 1 is 0 when x is 1: the sign in the bracket is the opposite.
Careful: not (x − 1): x = −1 ✗
So it crosses the x axis at x = 1 and x = 5: the points (1, 0) and (5, 0).
x axis : (1, 0) and (5, 0)
3 Vertex form: the lowest point
Vertex form: a bracket squared, plus a number.
y = (x − 3)2 − 4
A square is never below 0: a number times itself is never minus.
(x − 3) squared is smallest, 0, when x − 3 is 0: at x = 3.
(x − 3)² is never below 0; it is 0 when x = 3
There the square part is 0.
So y is just −4.
Anywhere else we add a bit more than 0, so y is bigger.
So this is the lowest point.
x = 3: y = 0 − 4 = −4
So the lowest point is (3, −4).
lowest point : (3, −4)
4 Show it on the graph
Here is the graph of y = x² − 6x + 5.
It crosses the y axis at (0, 5).
It crosses the x axis at 1 and 5.
The lowest point is (3, −4).
5 Check
Let's check that the three forms are the same.
Open the factored form: it gives x² − 6x + 5.
(x − 1)(x − 5) = (x² − 5x − x + 5) = x² − 6x + 5
Open the vertex form: (x − 3) squared is x² − 6x + 9.
Minus 4: x² − 6x + 5 too.
So it is right!
(x − 3)² − 4 = (x² − 6x + 9) − 4 = x² − 6x + 5
Let's look back.
First: standard form: the y axis.
Then: factored form: the x axis.
Then: vertex form: the lowest point.
Then: show it on the graph.
Last: check.
▶ Watch it solved
The question
Question: which form of y = x² − 6x + 5 shows where it crosses the x axis?
y = x2 − 6x + 5
1 Pick the form
A parabola can be written in three forms.
Each one shows one thing at a glance.
On the x axis, y is 0.
The factored form is a product, and a product is 0 when one bracket is 0.
So the factored form shows where it crosses the x axis.
standard: ax2 + bx + c → y axis at c
factored: a(x − r)(x − s) → x axis at r and s ✓
vertex: a(x − h)2 + k → turning point (h, k)
2 Write it in factored form
Factoring goes backwards.
We need two numbers.
They multiply to 5, and they add to −6.
They give the brackets.
? × ? = 5, ? + ? = −6
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
× is + → same signs: both −
We list the pairs that make 5, smallest first.
We add each pair.
We need 6.
1 and 5 give 6.
With the signs: −1 and −5.
1 × 5: 1 + 5 = 6 ✓
The two numbers go in the brackets: y = (x − 1)(x − 5).
That is the factored form.
y = (x − 1)(x − 5)
3 Read the x axis crossings
On the x axis, y is 0.
So the product must be 0.
on the x axis, y = 0: (x − 1)(x − 5) = 0
A product is 0 only when one of its parts is 0.
x − 1 is 0 when x = 1.
x − 5 is 0 when x = 5.
A × B = 0 → A = 0 or B = 0
x − 1 = 0 → x = 1
x − 5 = 0 → x = 5
Careful!
Not −1.
x − 1 is 0 when x is 1: the sign in the bracket is the opposite.
Careful: not (x − 1): x = −1 ✗
So it crosses the x axis at x = 1 and x = 5: the points (1, 0) and (5, 0).
points = (1, 0) and (5, 0)
4 Show it on the graph
Here is the graph of y = x² − 6x + 5.
It crosses the x axis at 1 and 5.
5 Check
Let's check in the question's own form, y = x² − 6x + 5.
x = 1 gives y = 0: on the x axis.
x = 1: y = 1² − 6 × 1 + 5 = 1 − 6 + 5 = 0
And x = 5 gives y = 0: on the x axis.
So it is right!
x = 5: y = 5² − 6 × 5 + 5 = 25 − 30 + 5 = 0
Let's look back.
First: pick the form.
Then: write it in factored form.
Then: read the x axis crossings.
Then: show it on the graph.
Last: check.
Answer points = (1, 0) and (5, 0)
▶ Watch it solved
The question
Question: which form of y = 2x² − 8x + 6 shows where it crosses the x axis?
y = 2x2 − 8x + 6
1 Pick the form
A parabola can be written in three forms.
Each one shows one thing at a glance.
On the x axis, y is 0.
The factored form is a product, and a product is 0 when one bracket is 0.
So the factored form shows where it crosses the x axis.
standard: ax2 + bx + c → y axis at c
factored: a(x − r)(x − s) → x axis at r and s ✓
vertex: a(x − h)2 + k → turning point (h, k)
2 Write it in factored form
First we take 2 out of every part: each part divided by 2.
Then x squared has no number in front.
2x² − 8x + 6: each part ÷ 2
So y is 2 times (x² − 4x + 3).
y = 2(x2 − 4x + 3)
Factoring goes backwards.
We need two numbers.
They multiply to 3, and they add to −4.
They give the brackets.
? × ? = 3, ? + ? = −4
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
× is + → same signs: both −
We list the pairs that make 3, smallest first.
We add each pair.
We need 4.
1 and 3 give 4.
With the signs: −1 and −3.
1 × 3: 1 + 3 = 4 ✓
The two numbers go in the brackets: y = 2(x − 1)(x − 3).
That is the factored form.
y = 2(x − 1)(x − 3)
3 Read the x axis crossings
On the x axis, y is 0.
So the product must be 0.
on the x axis, y = 0: 2(x − 1)(x − 3) = 0
A product is 0 only when one of its parts is 0.
The 2 in front is never 0.
x − 1 is 0 when x = 1.
x − 3 is 0 when x = 3.
A × B = 0 → A = 0 or B = 0
x − 1 = 0 → x = 1
x − 3 = 0 → x = 3
Careful!
Not −1.
x − 1 is 0 when x is 1: the sign in the bracket is the opposite.
Careful: not (x − 1): x = −1 ✗
So it crosses the x axis at x = 1 and x = 3: the points (1, 0) and (3, 0).
points = (1, 0) and (3, 0)
4 Show it on the graph
Here is the graph of y = 2x² − 8x + 6.
It crosses the x axis at 1 and 3.
5 Check
Let's check in the question's own form, y = 2x² − 8x + 6.
x = 1 gives y = 0: on the x axis.
x = 1: y = 2 × 1² − 8 × 1 + 6 = 2 − 8 + 6 = 0
And x = 3 gives y = 0: on the x axis.
So it is right!
x = 3: y = 2 × 3² − 8 × 3 + 6 = 18 − 24 + 6 = 0
Let's look back.
First: pick the form.
Then: write it in factored form.
Then: read the x axis crossings.
Then: show it on the graph.
Last: check.
Answer points = (1, 0) and (3, 0)
▶ Watch it solved
