What is the rectangular form of the parametric equations?
x(t)=t−2,y(t)=2t^2+4, where t is on the interval [−3,1].
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y =[ ] , where x is on the interval [ , ].
x = t - 2
y = 2t2 + 4
Let's solve the first equation for t and substitute that in for t into the second equation.
x = t - 2 Add 2 to both sides of this equation.
x + 2 = t
Since t = x + 2 we can substitute x + 2 in for t into the second equation.
y = 2t2 + 4
y = 2(x + 2)2 + 4
I'll let you expand the right side of that equation if necessary.
To find the smallest possible value of x, plug in the smallest possible value of t into the equation x = t - 2
And to find the biggest possible value of x, plug in the biggest possible value of t into the equation x = t - 2
When t = -3, x = -3 - 2 = -5
When t = 1, x = 1 - 2 = -1
So x is in the interval [-5, -1]
Here's a graph: https://www.desmos.com/calculator/ajymj9t4ve
x = t - 2
y = 2t2 + 4
Let's solve the first equation for t and substitute that in for t into the second equation.
x = t - 2 Add 2 to both sides of this equation.
x + 2 = t
Since t = x + 2 we can substitute x + 2 in for t into the second equation.
y = 2t2 + 4
y = 2(x + 2)2 + 4
I'll let you expand the right side of that equation if necessary.
To find the smallest possible value of x, plug in the smallest possible value of t into the equation x = t - 2
And to find the biggest possible value of x, plug in the biggest possible value of t into the equation x = t - 2
When t = -3, x = -3 - 2 = -5
When t = 1, x = 1 - 2 = -1
So x is in the interval [-5, -1]
Here's a graph: https://www.desmos.com/calculator/ajymj9t4ve