Given x = 3 + 2t
y = 2 - t
The given equations are parametric equations with parameter t.
Example: Find parametric equations of the line through (-1,3) and (1,1)
Example: Find a vector equation of the line through (-1,3) and (1,1).
Solve the first equation for t and plug
value into the second equation.
t2 = 3 - x
t = \/(3 - x)
y = \/(3 - x) - 1
(b) Write the parametric equations as a rectangular equation
x = 2 cos q
y = 5 sin q
+ sin2q = 1, solve for sin
in the above equations and substitute
into the identity statement.
x/2 = cos q
y/5 = sin q
+ sin2q = 1
(x/2)2 + (y/5)2 = 1
x2 + y2 = 1
We have an ellipse.
(c) Find a set of parametric equations to represent the graph
of y = x2 + 2x – 1
Let x = t
y = t2 + 2t - 1
(d) Find a set of parametric equations to represent the graph
of x2 + y2 = 4.
Divide by 4, x2 + y2 = 1
Considering cos2q + sin2q = 1, cos q = x/2 and sin q = x/2
so x = 2 cos q
y = 2 sin q
(e) A baseball is hit when it is 3 feet above
ground and leaves the bat with initial velocity of
150 ft per second and at an angle of elevation of 20o. A 6 mph wind is blowing in the
horizontal direction against the batter. A 20 ft high fence is 400 ft from home plate. Will the
hit be a home run?
Change 6 mph to ft/sec:
6 mi 1
1 min 5280 ft = 8.8
hr 60 min 60 sec mi sec
Set up the situation:
x represents the horizontal motion: x = 150t cos 20 o - 8.8t (distance obtained by the
horizontal velocity and windspeed)
y represents the vertical motion: y = -16t2 + 150t sin 20o + 3 (distance obtained by
gravity, vertical velocity, and original height)
The question now is, how high is the ball when it reaches the fence 400 ft out? Set x = 400
and solve for t.
400 = 150t cos 20 o - 8.8t
400 = (150 cos 20 o - 8.8)t
400 = t
150 cos 20 o - 8.8
3.027 = t
put this value for t
into the y equation to
see how the ball was at
y = -16( 3.027)2 + 150( 3.027) sin 20o + 3 = 11.701 ft
No, the hit was not a home run. It didn't make it over the fence.
in parametric mode in the calculator, set X1 = 400 and
Y1 = 20 t / tmax to show the fence. Then X2 and Y2 are the equations of