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Precalculus Pre-AP SUMMER 2015

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Page 1: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Precalculus Pre-APSUMMER 2015

Page 2: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Defining a Function

When defining a function, we use coordinate points… ( x , y )

Typically, we use “x” to find out the value of “y”, and vice versa.

Page 3: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Defining a Function…Parametrically

Up until this point, we’ve used equations to tell us the values of x and y.

Today, we’ll examine what it would look like if x and y were defined by equations using a THIRD variable t. This is called a parameter.

Page 4: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Parametric Equations

Why…

Some functions and graphs can best be defined parametrically, rather than only using x and y.

Page 5: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Parametric Equations

ExampleX and Y are defined by the equations…

x = t + 1

y = t2 + 2t

Find the points determined by…

t = -3, -2, -1, 0, 1, 2, and 3.

Page 6: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Parametric Equations

ExampleX and Y are defined by the equations…

x = t + 1

y = t2 + 2t

Find an algebraic relationship between x and y. This is often called (“eliminating the parameter”).

Is this relationship a function?

Page 7: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Parametric Equations

Pg. 135# 1-8

Page 8: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Applications of Parametric Equations

Remember projectile motion from Algebra 2…

y = -16t2 + v0t + y0

Page 9: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Applications of Parametric Equations

Remember projectile motion from Algebra 2…

y = -16t2 + v0t + y0

This was okay, but what DIDN’T we account for with this equation?

Page 10: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Here is how we’ll think about it now…

X represents horizontal motion.

Y represents vertical motion.

Page 11: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

is the angle made with the horizontal

is the gravity constant

is the initial height

Page 12: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Example 1

A golf ball is hit at an angle of 65o with an initial velocity of 71 feet per second.

Write a set of parametric equations to represent this situation.

What is the maximum height of the ball?

Page 13: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Example 1

A golf ball is hit at an angle of 65o with an initial velocity of 71 feet per second.

How long was the ball in the air?

How far has the ball traveled horizontally before it hits the ground?

Page 14: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Example 1

A golf ball is hit at an angle of 65o with an initial velocity of 71 feet per second.

How high was the ball after 1.1 seconds?

Page 15: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Example 2

A rock is thrown off of a 121 meter cliff at an angle of 43 degrees. The initial velocity is 28 meters per second.

Write a set of parametric equations to represent this situation.

What is the maximum height of the rock?

Page 16: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Example 2

A rock is thrown off of a 121 meter cliff at an angle of 43 degrees. The initial velocity is 28 meters per second.

How long was the rock in the air?

How far did the rock travel horizontally before it hit the ground?

Page 17: Precalculus Pre-AP SUMMER 2015. Defining a Function When defining a function, we use coordinate points… ( x, y ) Typically, we use “x” to find out

Projectile Motion… Version 2.0!!!

Example 2

A rock is thrown off of a 121 meter cliff at an angle of 43 degrees. The initial velocity is 28 meters per second.

How high was the ball after 0.6 seconds?

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