Find a cartesian equation for the following parametric graph. Indicate the portionof the graph traced by the parametrization. x = sec2 (t) -1, y = tan(t), -π/2 < t < π/2.
Question
Find a cartesian equation for the following parametric graph. Indicate the portionof the graph traced by the parametrization. x = sec2 (t) -1, y = tan(t), -π/2 < t < π/2.
Solution
To find a Cartesian equation for the parametric graph, we need to eliminate the parameter t.
Given the parametric equations:
x = sec²(t) - 1 y = tan(t)
We know that sec²(t) = 1 + tan²(t) from the Pythagorean identity in trigonometry.
So, we can substitute the equation for x into this identity to get:
x + 1 = 1 + tan²(t) x = tan²(t)
Since y = tan(t), we can substitute y into the equation to get:
x = y²
This is the Cartesian equation for the parametric graph.
The portion of the graph traced by the parametrization -π/2 < t < π/2 is the right half of the parabola, because tan(t) and sec(t) are undefined at t = π/2 and t = -π/2. Therefore, the graph does not include the points where x = -1.
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