sketch the curve using the parametric equation to plot the points. use an arrow to indicate the direction the curve is traced as t increases. Find the lenghth of the curve for o<t<1. Find an equation for the line tangent to the curve at the point where t=-t.

the equation for the parametric curve is defined by x=3-2t^2 and y=5-2t

To sketch the curve using the parametric equation, we need to plot the points for various values of t.

1. Choose several values for t within the given range (0 < t < 1). Let's use t = 0, 0.25, 0.5, 0.75, and 1.
For t = 0: x = 3 - 2(0)^2 = 3, y = 5 - 2(0) = 5. So the point (3, 5) belongs to the curve.
For t = 0.25: x = 3 - 2(0.25)^2 = 2.9375, y = 5 - 2(0.25) = 4.5. So the point (2.9375, 4.5) belongs to the curve.
Continue this process for the other values of t.

2. Once you have the points plotted, connect them in the order they are listed. Use an arrow to indicate the direction in which the curve is traced as t increases. This is usually from the starting point to the ending point.

Now let's find the length of the curve for 0 < t < 1.

3. To find the length of a parametric curve, we will use the formula for arc length:
s = ∫(t1 to t2) √[(dx/dt)^2 + (dy/dt)^2] dt.

4. Differentiating the parametric equations with respect to t, we get:
dx/dt = -4t
dy/dt = -2

5. Substituting these values into the arc length formula, we have:
s = ∫(0 to 1) √[(-4t)^2 + (-2)^2] dt

6. Simplifying the integral, we have:
s = ∫(0 to 1) √(16t^2 + 4) dt

7. Evaluating this integral, we find the length of the curve for 0 < t < 1.

To find the equation for the tangent line at the point where t = -t:

8. Substitute t = -t into the parametric equations:
x = 3 - 2(-t)^2 = 3 - 2t^2
y = 5 - 2(-t) = 5 + 2t

9. Differentiate x and y with respect to t to find the derivatives dx/dt and dy/dt, respectively.

10. Evaluate these derivatives at t = -t to find the slope of the tangent line at that point.

11. Use the slope and the point (3 - 2t^2, 5 + 2t) to write the equation of the tangent line in the form y = mx + b, where m is the slope and b is the y-intercept.