sketch the curve given by each pair of parametric equations over the given interval

Question

https://ibb.co/zh5fLw6

photo of problem ^

Answer 1

First, eliminate the parametric variable:

x = t² + 1

y = t - 6

It will be easier to start with y.

t = y + 6

Then, substitute this value for t in the other equation.

x = (y+6)² + 1 = ( y² + 12y + 36 ) + 1 = y² + 12y + 37

This can be graphed in Desmos over the interval [-4,4]. You can also graph the original parametric equations in Desmos to confirm that they are identical.

You can also set up a table using several sample values.

Answer 2

To sketch x = t² +1, y = t - 6 over the interval -4 ≤ t ≤ 4, it may be helpful to complete a table of ordered pairs.

This being a parametric set of equations for the curve we might use three columns in the table, one for t, one for x, and one for y. You can create rows in the table for t = -4, -3, -2, ... ,etc. up to t = 3, 4

Your initial table should look something like this.

t | x | y

-4 | |

-3 | |

-2 | |

-1 | |

0 | |

1 | |

2 | |

3 | |

4 | |

To complete the table for each value of t, plug them into the equations for x and y.

For example for t = -4, we get x = (-4)²+1 = 16 + 1 = 17 and y = (-4) - 6 = -10

So the first row of our table would be complete as below.

t | x | y

-4 | 17 | -10

-3 | |

-2 | | etc.

...

After completing all the rows of the table you can sketch the (x,y) points on the same coordinate plane and then have a rough idea of the overall graph of the parametric equations.

If I can help explain some more steps, please let me know. I'll be glad to help. Thank you!