Question

What does the graph of the parametric equations x(t)=2−t and y(t)=(t+3)^2, where t is on the interval [−4,0], look like?

Answer 1

see attached

Explanation:

You want the graph of (x, y) = (2 -t, (t+3)²) on the interval t ∈ [-4, 0].

Graph

You can use the parametric equations to find several points on the graph, or you can write y in terms of x:

x = 2 -t

t = 2 -x . . . . . . . . . add t-x to both sides

Substituting into the expression for y, we have ...

y = (t +3)²

y = (2 -x +3)²

y = (5 -x)² = (x -5)²

The graph of this is the parent parabola function y=x² shifted right by 5 units.

Domain

The domain of t is [-4, 0], so the domain of x is 2 -[-4, 0] = [6, 2]. The above-described parabola will be found between x=2 and x=6, on the interval [2, 6].

Range

The range is from the minimum at (5, 0) to the maximum at (2, 9). The range is described by the interval [0, 9].

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Additional comment

You know that x² ≡ (-x)², so we can write y = (5 -x)² in the form y = (x -5)² without changing any of the ordered pairs it describes. Having a positive x-coefficient tends to reduce anxiety (and errors), so the latter form is usually preferred.