Applications of the vertex formula

SAT Problem of the day: The graph of a quadratic function y is shown. For what value of x does y attain its greatest value?

y = – x2 + 6x – 3

Applications of the Vertex FormulaObjective (4.3): Develop and apply a method for finding the maximum height of a projectile.

(a) 0.5

(b) 3

(c) 5.5

(d) 6

(e) 8

• A projectile is an object in which the only force acting upon it is gravity.

Vocabularyre-visited

• A parabola is the graph of a quadratic function.

Which of the following would NOT be considered a projectile?

(a) Dropping a pumpkin off of a building.

(b) A plane taking off and then landing.

(c) Driving a golf ball.

(d) A person jumping on trampoline.

(b) A plane taking off and then landing.

Vertex

Vertex

The vertex of a parabola is either the lowest point on the graph or the highest point on the graph.

book page 276

minimum

maximum

When a parabola opens up and the vertexis the lowest point the y-coordinate of thevertex is the minimum.

When a parabola opens down and thevertex is the highest point the y-coordinateof the vertex is the maximum.

book page 277

When a parabola opens up its lowest point is known as the:

(a) minimum

(b) maximum

(c) vertex

(d) a & c

(e) b & c

(f) all of the above

(d) a & c

To help identify the vertex of a quadratic function we can use the following formula:

– b2a

x =

Where do we see quadratic functions in our everyday lives?

pollanywhere.com

Where do we see quadratic functions in real life?

What component do we often neglect when applying formulas for projectile motion?

(a) initial height

(b) initial velocity

(c) height

(d) air resistance

(e) velocity

(d) air resistance

Consider a firework display.

Collins Writing Type I:

When a projectile is released into the air a number of factors come into play including initial height, maximum height, time, and velocity. If you were designing a firework display why do you think each of these factors would be important?

Time: 90 seconds Length: 3 Lines

http://www.online-stopwatch.com/large-stopwatch/

h = –16t2 + v0t + h0

h

h0

t

v0

Initial Height

Height

Time

Initial Velocity (or speed)

Term used to represent the

earth’s gravity.

When a projectile is released into the air, what types of factors come into play?

The path of a firework can be modeled using a quadratic function

h = –16t2 + v0t + h0

– b2a

t =

We can use the vertex formula to determine the time it takes for a firework to explode, and the maximum height that it reaches.

On July 4th Ocean City has a firework display. The fireworks are ignited from the football field with an initial velocity of 96 feet per second.

How long does it take for the fireworks to reach their maximum height?

What is the maximum height reached by the fireworks?

initial velocity of 96

How long

maximum height

h = –16t2 + v0t + h0

– b2a

t

=

the football field

A professional pyro-technician shoots fireworks vertically into the air off of a building that is 80 feet tall. The initial velocity of the firework is 64 feet per second.

When will the fireworks reach their maximum height?

What is the maximum height reached by the fireworks?

initial velocity

When

maximum height

h = –16t2 + v0t + h0

– b2a

t

=

64 feet per second

Projectile MotionA.asf

80 feet tall

Textbook page 313Numbers 49 & 50

A baseball is thrown upward with an initial velocity of 48 feet per second from 6 feet above the ground. Determine the maximum height of the ball.

initial velocity

maximum height

h = –16t2 + v0t + h0

– b2a

t

=

6 feet above the groundof 48 feet per second

EXIT TICKET

– b2a

t =

h = –16t2 + v0t + h0