To sketch the curve defined by the parametric equations x = t² and y = t⁷, we first need to calculate several points by substituting various values of t.
Let’s choose values for t in a range to get a sense of how the curve behaves:
- For t = -2:
- x = (-2)² = 4
- y = (-2)⁷ = -128
- For t = -1:
- x = (-1)² = 1
- y = (-1)⁷ = -1
- For t = 0:
- x = 0² = 0
- y = 0⁷ = 0
- For t = 1:
- x = 1² = 1
- y = 1⁷ = 1
- For t = 2:
- x = 2² = 4
- y = 2⁷ = 128
Now we have the following points to plot:
- (4, -128)
- (1, -1)
- (0, 0)
- (1, 1)
- (4, 128)
To visualize the curve:
- Start plotting points on a Cartesian plane based on the calculated coordinates.
- Notice that as t increases or decreases, the curve appears to be steeper and changes direction sharply due to the higher power of y = t⁷ compared to x = t².
- Connecting the plotted points will show the overall shape of the curve. It will rise steeply in the positive and negative directions of y as x remains relatively small.
This gives a general shape of the curve, which is symmetric concerning the y axis due to the even power of x and the odd power of y.