Calculus HW Help

Question

Consider the parametric equations below.

x = t² − 4,    y = t + 3,    −3 ≤ t ≤ 3

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.

(b) Eliminate the parameter to find a Cartesian equation of the curve for 0 ≤ y ≤ 6

Answer 1

x = t^2 -4

y = t +3

t= y-3

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

x = y^2 -6y +5 = (y-5)(y-1) is the equation of x in terms of y, a rightward or leftward opening parabola y intercepts (0,5) and (0,1) and axis of symmetry y = 3, vertex = (-4,3) or (4,3) and x intercept = 5 or -5

as t goes from -3 to 3, x goes from t^2 -4, 9-4 = 5 to 5 and y goes from 0 to 6