To sketch the curve defined by the parametric equations x = t2 and y = t9, we can start by plotting several points by substituting different values of t.
Let’s choose a range of t values, both positive and negative, to see how the curve behaves:
- For t = -2:
x = (-2)2 = 4, y = (-2)9 = -512
Point: (4, -512) - For t = -1:
x = (-1)2 = 1, y = (-1)9 = -1
Point: (1, -1) - For t = 0:
x = (0)2 = 0, y = (0)9 = 0
Point: (0, 0) - For t = 1:
x = (1)2 = 1, y = (1)9 = 1
Point: (1, 1) - For t = 2:
x = (2)2 = 4, y = (2)9 = 512
Point: (4, 512)
Now, let’s observe the behavior of the curve:
- For large negative values of t, y decreases rapidly because of the exponent 9, while x increases positively.
- For values of t around 0, both x and y are at the origin (0,0).
- As t increases, x increases at a steady rate since it’s squared, while y increases much more steeply due to the exponent of 9.
After plotting these points, you will see a curve that is symmetric about the y-axis due to the even power in the x equation (t2). The behavior of the curve outside the y-axis is significantly steep because of the higher degree in the y equation. This results in a graph that grows rapidly for both positive and negative values of t.
In conclusion, sketching the curve defined by these parametric equations provides insight into its symmetry and steepness in different quadrants. The key is the careful plotting of points based on chosen values of t.