Sketch the plane curve represented by the given parametric equations. then use interval notation to give the relation's domain and range. x = 2t, y = t2 + t + 3

Domain: and Range:

Step-by-step explanation:

We have the parametric equations and .

Now, we will find the values of 'x' and 'y' for different values of 't'.

t                       :   -3       -2.5     -2     -1.5     -1     0       1      1.5      2

            :   -6       -5       -4       -3      -2    0      2      3        4

 :    9       6.75     5     3.75     3    3      5     6.75     9

Now, we can see that these parametric equations represents a parabola.

The general form of the parabola is .

Now, we have the point ( x,y ) = ( 0,3 ). This gives that c = 3.

Also, we have the points ( x,y ) = ( -2,3 ) and ( 2,5 ). Substituting these in the general form gives us,

4a - 2b + 3 = 3 → 4a - 2b = 0 → b = 2a.

4a + 2b + 3 = 5 → 4a + 2b = 2 → 2a + b = 1 →  2a + 2a = 1 ( As, b = 2a ) → 4a = 1 → .

So, .

Therefore, the equation of the parabola obtained is .

The graph of this function is given below and we can see from the graph that domain contains all real numbers and the range is .

Hence, in the interval form we get,

Domain is and Range is

Domain:

Range:

Step-by-step explanation:

we are given parametric equation as

We can change into rectangular equation

we can eliminate t from first equation and plug into second equation

now, we can plug that into second equation

now, we can draw graph

Domain:

we know that

domain is all possible values of x for which any function is defined

we can see that our equation is parabolic

and it is defined for all values of x

so, domain will be

Range:

we know that

range is all possible values of y

we can see that

smallest y-value is 2.75

so, range will be

what's that