Worked examples
The question
Question: graph y = x².
y = x2
x² means 1x², so a = 1.
a is positive, so the U opens up.
a = 1 > 0 → opens up
1 Find the vertex
The vertex is the turning point of the curve.
Its x is minus b/2a.
In y = ax² + bx + c, here a = 1 and b = 0.
x = −b2a = −02 × 1
Minus 0 is 0.
2 times 1 is 2.
x = 02
0 divided by 2 is 0.
x = 0
To get y, put x = 0 in.
We get y = 0.
y = 02 = 0
So the vertex is (0, 0).
vertex = (0, 0)
2 Make a table
We make a table around the vertex: x from −2 to 2.
At x = 0, the vertex: y = 0.
0
Now the right side.
Put x = 1 in.
We get y = 1.
x = 1: y = 12 = 1
1
Next.
Put x = 2 in.
We get y = 4.
x = 2: y = 22 = 4
4
The curve is the same on both sides of the vertex, like a mirror.
So x = −1 gives the same y as x = 1.
1
And x = −2 is the same as x = 2.
4
3 Plot the points
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put the points from the table on the grid.
The mirror line goes through the vertex: x = 0.
4 Draw the curve
Join the points with a smooth curve, not straight lines.
This U shape is a parabola.
5 Crossing the x-axis
The curve just touches the x-axis at x = 0, at the vertex.
6 Check
Let's check the mirror.
Put x = −2 straight in.
We get 4, the same as in the table.
y = (−2)2 = 4
Let's look back.
First: find the vertex.
Then: make a table.
Then: plot the points.
Then: draw the curve.
Then: crossing the x-axis.
Last: check.
▶ Watch it solved
The question
Question: graph y = x² − 4x + 3.
y = x2 − 4x + 3
x² means 1x², so a = 1.
a is positive, so the U opens up.
a = 1 > 0 → opens up
1 Find the vertex
The vertex is the turning point of the curve.
Its x is minus b/2a.
In y = ax² + bx + c, here a = 1 and b = −4.
x = −b2a = −(−4)2 × 1
Minus a minus is plus: minus −4 is 4.
2 times 1 is 2.
x = 42
4 divided by 2 is 2.
x = 2
To get y, put x = 2 in.
We get y = −1.
y = 22 − 4(2) + 3
= 4 − 8 + 3 = −1
So the vertex is (2, −1).
vertex = (2, −1)
2 Make a table
We make a table around the vertex: x from 0 to 4.
At x = 2, the vertex: y = −1.
−1
Now the right side.
Put x = 3 in.
We get y = 0.
x = 3: y = 32 − 4(3) + 3
= 9 − 12 + 3 = 0
0
Next.
Put x = 4 in.
We get y = 3.
x = 4: y = 42 − 4(4) + 3
= 16 − 16 + 3 = 3
3
The curve is the same on both sides of the vertex, like a mirror.
So x = 1 gives the same y as x = 3.
0
And x = 0 is the same as x = 4.
3
3 Plot the points
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put the points from the table on the grid.
The mirror line goes through the vertex: x = 2.
4 Draw the curve
Join the points with a smooth curve, not straight lines.
This U shape is a parabola.
5 Crossing the x-axis
The curve crosses the x-axis at x = 1 and x = 3.
There, y is 0.
6 Check
Let's check the mirror.
Put x = 0 straight in.
We get 3, the same as in the table.
y = 02 − 4(0) + 3
= 0 + 0 + 3 = 3
Let's look back.
First: find the vertex.
Then: make a table.
Then: plot the points.
Then: draw the curve.
Then: crossing the x-axis.
Last: check.
▶ Watch it solved
The question
Question: graph y = 2x² − 4x − 1.
y = 2x2 − 4x − 1
The number in front of x² is a = 2.
a is positive, so the U opens up.
a = 2 > 0 → opens up
1 Find the vertex
The vertex is the turning point of the curve.
Its x is minus b/2a.
In y = ax² + bx + c, here a = 2 and b = −4.
x = −b2a = −(−4)2 × 2
Minus a minus is plus: minus −4 is 4.
2 times 2 is 4.
x = 44
4 divided by 4 is 1.
x = 1
To get y, put x = 1 in.
We get y = −3.
y = 2(1)2 − 4(1) − 1
= 2 − 4 − 1 = −3
So the vertex is (1, −3).
vertex = (1, −3)
2 Make a table
We make a table around the vertex: x from −1 to 3.
At x = 1, the vertex: y = −3.
−3
Now the right side.
Put x = 2 in.
We get y = −1.
x = 2: y = 2(2)2 − 4(2) − 1
= 8 − 8 − 1 = −1
−1
Next.
Put x = 3 in.
We get y = 5.
x = 3: y = 2(3)2 − 4(3) − 1
= 18 − 12 − 1 = 5
5
The curve is the same on both sides of the vertex, like a mirror.
So x = 0 gives the same y as x = 2.
−1
And x = −1 is the same as x = 3.
5
3 Plot the points
Draw the grid.
The x-axis goes across.
The y-axis goes up and down.
They meet at 0.
That point is called the origin.
Put the points from the table on the grid.
The mirror line goes through the vertex: x = 1.
4 Draw the curve
Join the points with a smooth curve, not straight lines.
This U shape is a parabola.
5 Crossing the x-axis
The curve crosses the x-axis between −1 and 0, and between 2 and 3.
6 Check
Let's check the mirror.
Put x = −1 straight in.
We get 5, the same as in the table.
y = 2(−1)2 − 4(−1) − 1
= 2 + 4 − 1 = 5
Let's look back.
First: find the vertex.
Then: make a table.
Then: plot the points.
Then: draw the curve.
Then: crossing the x-axis.
Last: check.
▶ Watch it solved
