1. Warm-Up 1/16 1. 2. C
Assignments Questions? H

3. Rigor: You will learn how to graph parametric equations and how to write equations in rectangular form. Relevance: You will be able to use parametric equations to solve real world problems related to the motion of projectiles.
MA.912. A.2.11

4. 7-5 Parametric Equations

6. The variable t is called a parameter.
The values of t trace the curve in a specific direction called the orientation.

7. Example 1: Sketch the curve given by each pair of parametric equations over the interval.
a. 𝑥= 𝑡 2 +5 and 𝑦= 𝑡 2 +4;−4≤𝑡≤4 t x y
– 4
– 3
– 2
– 1 1 2 3 4 21 2 14
2.5 9 3 6
3.5 5 4 6
4.5 9 5 14
5.5 21 6

8. Example 1: Sketch the curve given by each pair of parametric equations over the interval.
b. 𝑥= 𝑡 and 𝑦= 𝑡 4 +4;−8≤𝑡≤8 t x y
– 8
– 6
– 4
– 2 2 4 6 8 21 2 14
2.5 9 3 6
3.5 5 4 6
4.5 9 5 14
5.5 21 6

9. Example 1:

10. Example 2: Write the parametric equations in rectangular form.
𝑥=−3𝑡 and 𝑦= 𝑡 2 +2
𝑥=−3𝑡
𝑦= 𝑡 2 +2
𝑦= − 𝑥
𝑥 −3 =𝑡
𝑦= 1 9 𝑥 2 +2
𝑡=− 𝑥 3

11. Example 3: Write the parametric equations in rectangular form
Example 3: Write the parametric equations in rectangular form. then graph the equation. State domain restrictions.
𝑥= 1 𝑡 and 𝑦= 𝑡+1 𝑡
𝑦= 𝑥 2 +1
𝑦= 1 𝑥 𝑥 2
𝑦= 𝑡+1 𝑡
𝑥= 1 𝑡
𝑥 𝑡 =1
𝑦= 1 𝑥 𝑥 2 𝑥 𝑥 2
𝑡 = 1 𝑥
𝑡 2 = 1 𝑥 2
𝑦= 1+ 𝑥 2 𝑥 𝑥 2
𝑡= 1 𝑥 2
D: x >0
𝑦= 1+ 𝑥 2 1
𝑦=1+ 𝑥 2

12. Example 5: Use the parameter to write the parametric equations that can represent 𝑦= 𝑥 2 −4. Then graph the equation, indicating the speed and orientation.
a. 𝑡=𝑥
𝑥=𝑡
𝑦= 𝑥 2 −4
𝑦= 𝑡 2 −4
𝑦= 𝑡 2 −4
b. 𝑡=4𝑥+1
𝑥= 𝑡−1 4
𝑥= 𝑡−1 4
𝑦= 𝑡− −4
𝑦= 𝑡 − 𝑡 8 − 63 16
𝑦= 𝑡 2 −2𝑡+1 16 − 64 16
𝑦= 𝑡 2 −2𝑡−63 16

13. −1 math! 7-5 Assignment: TX p469, 2-16 Even & 26-36 Even
only graph 2-8 not 32
Chapter 7 Test Thursday 1/23
You will need to know formulas.

14. Example 4: Write the parametric equations in rectangular form
Example 4: Write the parametric equations in rectangular form. then graph the equation.
𝑥=2 cos 𝜃 and 𝑦=4 sin 𝜃
cos 𝜃 = 𝑥 2
sin 𝜃 = 4 4
sin 2 𝜃 + cos 2 𝜃 =1
𝑥 𝑦 =1
𝑥 𝑦 =1