Parametric equations
Graph and convert parametric equations in Grade 10 precalculus by eliminating the parameter t to find rectangular form, and use them to model projectile motion and curves.
Key Concepts
When two variables, like $x$ and $y$, are both expressed in terms of a third variable, such as $t$, the equations used are called parametric equations. For example, the area a painter covers and her earnings can be described by the equations $x = 200t$ and $y = 30t$.
Example 1: A painter's progress can be tracked with area painted $x = 150t$ and earnings $y = 25t$ dollars. Example 2: A golf ball's flight path is modeled by its horizontal position $x = 100t$ and vertical position $y = 16t^2 + 80t$. Example 3: A puppy's growth is shown by its height $x = 1.5t + 12$ and its weight $y = 9t$.
Think of parametric equations as a dynamic duo! Instead of relating $x$ and $y$ directly, they both rely on a third 'manager' variable, usually time. This manager tells both $x$ and $y$ what to do at every moment, creating a path or a story that unfolds over time. It's the ultimate behind the scenes coordinator for motion and change.
Common Questions
What are parametric equations and what is the parameter?
Parametric equations express x and y as separate functions of a third variable t (the parameter). For example, x=t+1 and y=t²-1 define a curve traced as t varies.
How do you eliminate the parameter to get a rectangular equation?
Solve one parametric equation for t, then substitute into the other. For x=t+1 and y=t²-1: t=x-1, so y=(x-1)²-1 = x²-2x.
How are parametric equations used to model projectile motion?
Projectile position is modeled as x=v₀cos(θ)·t and y=v₀sin(θ)·t - (1/2)gt², where t is time, v₀ is launch speed, θ is angle, and g is gravitational acceleration.