Parametric Equations Unit 3

What are parametrics? Normally we define functions in terms of one variable – for example, y as a function of x. Suppose in a graph, each (x, y) depended on a third variable t (for instance – time). We would call t the parameter and call the equation a parametric equation.

Why use parametric equations? As an object moves, its position (vertical and horizontal) is often dependent on a third variable (time). Sometimes it is easier to write the equation parametrically. Easier to graph the inverse of a function Allows you to use your calculator to graph non- functions. Parametric equations define the path of the curve. You can put restrictions on t so you start and end the graph at a specific point. For example, you can set t values so you graph only a segment, not the entire line.

From the Video ExampleTXY TXY

EXAMPLE 1: Graph Graph this parametric equation: Fill in the table for t values [-5, 5] t xy

Using Your Calculator Press MODE Go to FUN and choose PAR Click enter and go to Y=. – There are two lines x= and y= for the parametric equations Use the X, T, θ,N button to get the T variable

Example 2: Inverses Suppose. How could you find the inverse of this? Problems: Solving for y when You would have to graph it one half at a time because its not a function.

Example 2, cont. Instead, define it as Then the inverse is simply (switch x and y) Graph them both.

Graphing the Inverse

Example 3: Graph a cloud Graph Settings: t values: x and y: [-10,10]

Projectile Motion Suppose you toss a ball in the air. Its position (ordered pairs (x, y)) is defined parametrically by this set of equations: v x is the horizontal velocity at release v y is the vertical velocity at release The point at which you release it is the ordered pair (d 0, h 0 ) g is the gravitational constant (9.8 in m/s 2 or 32 in ft/sec 2 )

Example 4: Basketball Shot In a basketball game, a free throw is released from a point 5 feet above the ground and 1 foot in front of the free throw line. The vertical velocity at release is 17 ft/sec and horizontal velocity is 15 ft/sec. (a) Give the equations for the ball’s horizontal distance x and vertical distance y at time t: (b) Will the ball go directly into the basket? Assume the basket is 10 feet high and 14 feet horizontally from the point of release. (c) Look at the info on the graph. What does x, y, and t tell us? Trace it.

Closure What are parametric equations? Why do we use parametric equations? Given the parametric equations representing the path of a projectile: What do the 15 and 13 represent? What does the 1.5 and the 7 represent? At time = 1, how high and how far is the projectile? Approximately how long does it take to hit the ground?