Answer:
Domain: [tex]( -\infty,\infty )[/tex] and Range: [tex][ -1,\infty )[/tex]
Step-by-step explanation:
We have the parametric equations [tex]x= 2t[/tex] and [tex]y=t^{2}+t+3[/tex].
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
[tex]x= 2t[/tex] : -6 -5 -4 -3 -2 0 2 3 4
[tex]y=t^{2}+t+3[/tex] : 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 [tex]y=ax^{2}+bx+c[/tex].
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 → [tex]a=\frac{1}{4}[/tex].
So, [tex]b=\frac{1}{2}[/tex].
Therefore, the equation of the parabola obtained is [tex]y=\frac{x^{2}}{4}+\frac{x}{2}+3[/tex].
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 [tex]y\geq -1[/tex].
Hence, in the interval form we get,
Domain is [tex]( -\infty,\infty )[/tex] and Range is [tex][ -1,\infty )[/tex]
Answer:
Domain:
[tex](-\infty,\infty)[/tex]
Range:
[tex][2.75,\infty)[/tex]
Step-by-step explanation:
we are given parametric equation as
[tex]x=2t[/tex]
[tex]y=t^2+t+3[/tex]
We can change into rectangular equation
we can eliminate t from first equation and plug into second equation
[tex]x=2t[/tex]
[tex]t=\frac{x}{2}[/tex]
now, we can plug that into second equation
[tex]y=(\frac{x}{2})^2+\frac{x}{2}+3[/tex]
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
[tex](-\infty,\infty)[/tex]
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
[tex][2.75,\infty)[/tex]
