Worked examples
The question
Question: graph and remove t from x = t + 1, y = 2t.
x = t + 1 and y = 2t
1 Make a table
Each t gives a point.
Start with t = −2.
x is (−2) + 1: −1.
t = −2: x = (−2) + 1 = −1
y is 2 × (−2): −4.
y = 2 × (−2) = −4
Put the t's in a table.
Work out x for each t,
−1
0
1
2
3
and y for each t.
−4
−2
0
2
4
2 Plot the points
Draw the grid.
Put each (x, y) on the grid.
3 Join in order of t
Join the points in order: t = −2, then −1, and on.
The arrows show the way t grows.
t can go on, so the curve goes on.
4 Remove t
To get t alone, start with x = t + 1.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
x − 1 = t + 1 − 1
So t = x − 1.
t = x − 1
Put t into y = 2t, with its bracket.
y = 2(x − 1)
Careful!
2 multiplies all of t.
Keep the bracket, then open it.
Careful: not y = 2x − 1 ✗
Open the bracket.
So y = 2x − 2: a straight line.
y = 2x − 2
5 Check
Let's check with a point from the table: t = 2 gave (3, 4).
Put x = 3 into the answer.
We get y = 4, the same as the table.
Right!
x = 3: y = 2x − 2 = 4
Let's look back.
First: make a table.
Then: plot the points.
Then: join in order of t.
Then: remove t.
Last: check.
▶ Watch it solved
The question
Question: graph and remove t from x = t + 1, y = t².
x = t + 1 and y = t2
1 Make a table
Each t gives a point.
Start with t = −2.
x is (−2) + 1: −1.
t = −2: x = (−2) + 1 = −1
y is (−2)²: 4.
Minus times minus is plus.
y = (−2)² = 4
Put the t's in a table.
Work out x for each t,
−1
0
1
2
3
and y for each t.
4
1
0
1
4
2 Plot the points
Draw the grid.
Put each (x, y) on the grid.
3 Join in order of t
Join the points in order: t = −2, then −1, and on.
The arrows show the way t grows.
t can go on, so the curve goes on.
4 Remove t
To get t alone, start with x = t + 1.
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
x − 1 = t + 1 − 1
So t = x − 1.
t = x − 1
Put t into y = t², with its bracket: y = (x − 1)².
y = (x − 1)2
Careful!
The whole of x − 1 is squared, not only the last number.
So it keeps its bracket.
Careful: not y = x − 1² ✗
5 Check
Let's check with a point from the table: t = 2 gave (3, 4).
Put x = 3 into the answer.
We get y = 4, the same as the table.
Right!
x = 3: y = (3 − 1)² = 4
Let's look back.
First: make a table.
Then: plot the points.
Then: join in order of t.
Then: remove t.
Last: check.
Answer Careful: not y = x − 1² ✗
▶ Watch it solved
The question
Question: graph and remove t from x = 3 cos t, y = 3 sin t.
x = 3 cos t and y = 3 sin t
1 Make a table
At 0°, 90°, 180°, 270° and 360°, cos and sin are easy: 1, 0 or −1.
They are the x and y of a point going round a circle of radius 1.
cos: 1, 0, −1, 0, 1 sin: 0, 1, 0, −1, 0
Take t = 0°, 90°, 180°, 270° and 360°.
x is 3 times cos t,
3
0
−3
0
3
and y is 3 times sin t.
0
3
0
−3
0
2 Plot the points
Draw the grid.
Put each (x, y) on the grid.
At 360° we are back at the start.
3 Join in order of t
Join them in order of t: it goes round, against the clock.
A circle.
4 Remove t
For every t: cos² t + sin² t = 1.
It is Pythagoras on the circle of radius 1: across cos t, up sin t, long side 1.
cos²t + sin²t = 1
Get cos t alone: divide x by 3.
x3 = cos t
And sin t: divide y by 3.
y3 = sin t
Square both and add them.
(x3)2 + (y3)2 = cos²t + sin²t
Careful!
Not cos t plus sin t: that is not always 1.
Only the squares add up to 1.
Careful: square first, then add
That is 1, for every t.
So t is gone.
(x3)2 + (y3)2 = 1
Each square is 9 at the bottom.
Multiply both sides by 9: x² + y² = 9.
A circle of radius 3.
x2 + y2 = 9
× 9: x²9 × 9 = x²
5 Check
Let's check with t = 90°: the point (0, 3).
Put it in the answer: we get 9, the right side.
Right!
0² + 3² = 9
Let's look back.
First: make a table.
Then: plot the points.
Then: join in order of t.
Then: remove t.
Last: check.
▶ Watch it solved
