2.4 Parametric form

The figure shows the graph of a curve. Which one is the correct parametric form of the curve?

Let [math]C_1: \begin{cases}x=8+t\\y=12+t\end{cases}\quad t\in\mathbb{R}[/math] and [math]C_2:\begin{cases}x=-1+2t\\y=15-6t\end{cases}\quad t\in\mathbb{R}[/math] be two curves on the [math]xy[/math]-plane. Find the [math]y[/math]-coordinate of the intersection of [math]C_1[/math] and [math]C_2[/math].

Find the [math]y[/math]-intercept(s) of the curve [math]C: \begin{cases}x=t^2-4\\y=3^t-1\end{cases}\quad t\in[-3,1][/math].

Let [math]C: 4x+8=(y-1)^2[/math] be a curve where [math]y>0[/math]. A particle [math]P[/math] is on [math]C[/math]. If [math]P[/math] is in the second quadrant when [math]a<t<\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers, what are [math]a[/math] and [math]b[/math]?

Mr. Donkey wants to rewrite a curve [math]C: x^2+2y^2=3[/math] where [math]x\geq0[/math]. His work is shown below:

Let [math]x=t[/math]. The curve can be written as [math]y^2=\dfrac{3-t^2}{2}[/math]. Therefore, [math]y=\pm\sqrt{\dfrac{3-t^2}{2}}[/math].

Is his work correct?

Consider a point [math]A(0,7)[/math] and a line [math]L: 2x-y=3[/math]. [math]B[/math] is a point on [math]L[/math].

Consider the parametric form [math]\begin{cases}x=\sqrt{t-4}\\y=2(t-4)+3\end{cases}[/math]

Let [math]C:\begin{cases}x=t^2-1\\y=2t+3\end{cases}, t\in\mathbb{R}[/math] be a curve.

(a) Find the summation of the [math]y[/math]-intercepts of [math]C[/math].

(b) Let [math]\ell: y=-2x+13[/math] be a line. [math]C[/math] and [math]\ell[/math] intersect at [math](a,b)[/math] where [math]a, b[/math] are constants and [math]ab<0[/math]. Find [math]a[/math] and [math]b[/math].

Let [math]f(x)[/math] be a function. If [math]f(x)[/math] can be written in parametric form [math]\begin{cases}x=t^2\\y=\sqrt{t-2}+1\end{cases}[/math], find the domain and range of [math]f(x)[/math].

(a) If the domain of [math]f(x)[/math] is [math][a,\infty)[/math] where [math]a[/math] is a constant, what is the value of [math]a[/math]?

(b) If the range of [math]f(x)[/math] is [math][b,\infty)[/math] where [math]b[/math] is a constant, what is the value of [math]b[/math]?

While defending Winterfell during its siege, Tyrion Lannister catapulted his secret weapon wildfire, at the attacking fleet. One of the archers on a boat under fire noticed a second after the catapult launched the projectile, that it was approaching following the parametric equation:
[math]\begin{cases}
x=230-10t\\
y=20+33t-5t^2
\end{cases}[/math]
The archer does some quick calculating and launches his arrow is the air in an attempt to intercept the projectile before it lands on his boat. His dart follows the path:
[math]\begin{cases}
x=45(t-1)\\
y=35(t-1)-5(t-1)^2
\end{cases}[/math]
Does his arrow strike the projectile and save this ship? Write “Yes” or “No”

If [math]x=t^2, y=t[/math] where [math]t\geq0[/math], then which relationship between [math]x[/math] and [math]y[/math]?