SME
Grade 12 Math · Precalculus
Parametric equations
Use this for “x = 1 + 2t and y = 3 − t. Where is the point at t = 2, and what is the slope of the line?”
Diagram
At t = 2 the point is (5, 1).
The arrows show the way t runs.
This is an example. Type your own values to start a new problem.
x
=
x₀
+
a
t
y
=
y₀
+
(
b
)
t
t
Parameter
example
m
Slope of the line
worked out (not typed)
Formulas
x
=
x₀
+
a
×
t
5 = 1 + 2 × 2
y
=
y₀
+
b
×
t
1 = 3 + (−1) × 2
m
=
b
a
−0.5
=
−
1
2
Assumptions
•
t is the input; x and y are both outputs, and the point (x, y) moves as t grows.
•
The arrows show the direction the point moves as t increases.
•
To eliminate t, solve one equation for t and put it into the other: here a line of slope b ÷ a.
Step-by-step
WE KNOW
t = 2, x₀ = 1, y₀ = 3, a = 2, b = −1
FIND
x at t (x), y at t (y), slope of the line (m)
Step 1 · Find x at t (x)
Use x = x₀ + a × t
Start
at
x₀
and
move
a
for
each
1
of
t
.
x
=
1
+
2
×
2
x
=
1
+
4
x
=
5
Step 2 · Find y at t (y)
Use y = y₀ + b × t
Start
at
y₀
and
move
b
for
each
1
of
t
.
y
=
3
−
1
×
2
y
=
3
−
2
y
=
1
Step 3 · Find slope of the line (m)
Use m = b ÷ a
Each
1
of
t
moves
a
across
and
b
up,
so
the
path’s
slope
is
b
over
a
.
m
=
−
1
2
m
=
−0.5
→
y
−
3
=
−0.5(
x
−
1)
Check
Put the numbers back in. Both sides should match.
5 = 1 + 2 × 2 ✓
1 = 3 − 1 × 2 ✓
−0.5
=
−
1
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.