Quadratic Models



Quadratic Models

Note: Much of the material from this first section is from Contemporary Mathematics in Context 2A p265f

For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as [pic].

a. What sort of graph would you expect for the (time in flight, height) relation? Sketch to the right.

b. How could you use the given rule relating height to time in flight to find when the shot might reach the height of the basket (10 feet)?

c. How could you find the time when the ball would hit the floor, if it missed the basket entirely?

Equations in the form: are called quadratic models.

The distance in meters an object falls on earth (neglecting air-resistance) where t is time in seconds can be modeled with the equation:

a. How would this equation be different if we measured in feet instead of meters?

b. Would this equation be different if we measured in meters per second on the moon? Why?

Platform Diving: What would be your equation to find height above the water (h)?

Diving Estimates:

Time in Flight (t) Distance Fallen (d) Height Above Water (h)

0

0.25

0.50

0.75

1.00

1.25 1.50

1.75

2.00

How about heights other than 10 meters?

a. What would be your equation for height above the water if the platform were 20 meters high?

b. How could we estimate the time it would take a diver to hit the water from a 20m platform?

c. What would be the equation for height above the ground for a marble dropped from a 50m building?

d. What would be the equation for height of a pop fly starting from a maximum height of 24 meters?

Gravity pulls falling objects toward the earth. How does it affect objects that appear to be flying upward such as a ball kicked up in the air?

A ball kicked in the air, a bullet shot in the air, and a person jumping all have _______________________ (or initial speed).

Suppose a diver jumps off a 3 meter springboard, moving upward at a speed of 4 meters per second, into a world without gravity. How would her height increase over time?

Springboard Diving with No Gravity:

Time in Flight (seconds) 0 1 2 3 4 5 6

Height Above Water (meters)

a. What would the equation be for this situation?

b. Is this situation quadratic or linear?

c. Where do we see the diver’s initial speed in this equation?

d. What would the equation be if the diver’s initial speed was 2.5 meters per second?

e. What would the equation be if the diver’s initial speed was still 4 meters per second, but she jumped from a 5 meter board instead?

Let’s put gravity back into our diver’s world. What would be the equation modeling the height of a diver jumping from a 3 meter springboard with an initial velocity of 4 meters per second upward?

The height of a football t seconds after a punt depends upon the initial height and velocity of the ball and on the downward pull of gravity. Suppose a punt leaves the kicker’s foot at an initial height of 0.8 meters with initial upward velocity of 20 meters per second. What is the equation that would model this situation?

Example 1 – Are these linear or quadratic? Identify each term.

A. f(x) = (2x + 3)(x – 4) B. f(x) = 3(x2 – 2x) – 3(x2 – 2) C. f(x) = (x – 5)(3x – 1)

A ______________ is the graph of a quadratic function.

When a>0, the parabola opens _________. When a ................
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