Visualizing Quadratic Functions

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• Visualizing Quadratic Functions
• John Peterson
• Yale University
• Languages for Mathematics
• Education
• www.haskell.org/edsl

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Turning Math into Pictures

• I study languages that allow people to
• use computers in new ways.
• Interactive pictures give us a way of
• showing the underlying intuition behind
• the mathematics.

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Graphing a Parabola

• Run 01-basicparabola.pan

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What did we see?

• What does a do?
• What does b do?
• What does c do?
• Which controls affect the axis of symmetry?
• What is the symmetry of b?
• What is the path of the minima as b changes?

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Why Parabolas?

• Parabolas describe an important natural
  phenomena the motion of bodies
• This was first noted by Galileo

From Galileos notes
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Gravity

• Gravity causes the downward velocity of an object
  to increase linearly.
• Time Velocity
• 0 0
• 1 1
• 2 2
• This assumes the gravitational constant is 1

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Turning Velocity into Positions

• If we know how fast something is moving
• how do we convert this to distance
• when the speed is not constant?

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Turning Velocity into Positions

• If we know how fast something is moving
• how do we convert this to distance
• when the speed is not constant?
• Well approximate the speed pretend its
  constant for the duration of a short interval of
  time

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Watch a falling body

• Run 02-fallingbody
• (Why dont things like golf balls travel in real
  parabolic paths?)

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Taking Apart the Parabola

• To understand why a, b, and c do what they do we
  will visualize all parts of the
• parabola the line bx c
• and the parabola ax
• Run 03-compositeparabola

2
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Understanding a, b, and c

• Observations
• a determines the curvature of the parabola
• b determines the slope of the line where it
  crosses x0
• c determines the y intercept

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Throwing Stuff

• One of the oldest math problems is how
• to fire a projectile the farthest.
• The initial velocity is constant we seek the
  best angle (slope) to launch at.
• Run 04-throw to see how this works
• What initial slope is best?

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Shooting the Farthest

• Distance traveled is proportional to the product
  of the upward and forward velocity.
• The square of the total velocity is the sum of
  the squares of the forward and upward velocity
• A graph of this relationship is itself a
  parabola!
• The maximum of this parabola occurs where the
  forward and upward velocity are equal (45 degrees)

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Solving Quadratic Equations

• While you would normally approach this problem
  algebraically, you can also do this visually.
• Consider three coordinate system transformations
• Vertical Scaling f(x) / v
• Vertical Translation f(x) - a
• Horizontal Translation f(x - b)
• All three f(x - b) / v - a