Question

A curve is described by the following parametric equations:

x=3+t
y=t^2-4
A. The curve is a parabola with a vertex at (3,-4) and is traced from left to right for increasing values of t.
B. The curve is a parabola with a vertex at (3,-4) and is traced from right to left for increasing values of t.
C. The curve is a parabola with a vertex at (-3,4) and is traced from left to right for increasing values of t.
D. The curve is a parabola with a vertex at (-3,4) and is traced from right to left for increasing values of t.

Answer 1

Correct option: A

Explanation:

To find the equation of the curve related to x and y, we can isolate t in the first equation and then use the value of t in the second equation:

x = 3 + t -> t = x - 3

y = t^2 - 4 -> y = (x-3)^2 - 4 = x^2 - 6x + 9 - 4 = x^2 - 6x + 5

Looking at the curve, we know it is a parabola. To find its vertex, we can use the formula:

x_vertex = -b / 2a

x_vertex = 6 / 2 = 3

To find y_vertex, we use x = x_vertex in the equation:

y_vertex = 3^2 - 6*3 + 5 = -4

So the vertex is (3, -4)

Looking at the first equation, we can see that an increase in t causes an increase in x, so we know that the parabola is traced from left to right for increasing values of t.

Correct option: A