Answer:
See attachment
Step-by-step explanation:
The given parametric equations are;
[tex]x=2t[/tex] and [tex]y=t+5[/tex], [tex]-2\le t\le 3[/tex].
We can graph this by plotting some few points within the given range or eliminate the parameter to identify the type of curve.
Plotting points;
When [tex]t=-2[/tex],
[tex]x=2(-2)=4[/tex] and [tex]y=-2+5=3[/tex]
This gives the point (-4,3).
When [tex]t=0[/tex]
[tex]x=2(0)=0[/tex] and [tex]y=0+5=5[/tex]
This gives the point (0,5).
When [tex]t=3[/tex]
[tex]x=2(3)=6[/tex] and [tex]y=3+5=8[/tex]
This gives the point (6,8).
We plot these points and draw a straight line through them.
Eliminating the parameter.
[tex]x=2t[/tex]
[tex]y=t+5[/tex]
Make t the subject in the second equation;
[tex]t=y-5[/tex]
Substitute into the first equation;
[tex]x=2(y-5)[/tex]
This implies that;
[tex]x=2y-10[/tex]
[tex]y=\frac{1}{2}x+5[/tex]
This is an equation of a straight line with slope [tex]\frac{1}{2}[/tex] and y-intercept 5 on the interval
[tex]-4\le x \le 6[/tex]
