Parabolas An Introduction

1
Parabolas An Introduction

• by Gary Greer
• for Grade 10 Math

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Curves can model things we see

• Bridges
• A ball thrown in the air

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What is a parabola?
Axis of Symmetry
Vertex
Vertex
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Graphing a Parabola Table of Values

• x y x²
• -3 9
• -2 4
• -1 1
• 0 0
• 1 1
• 2 4
• 3 9

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Lets try another

• x y -x²
• -3 -9
• -2 -4
• -1 -1
• 0 0
• 1 -1
• 2 -4
• 3 -9

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How do you tell if it is a true parabola?

• Each true parabola has a pattern
• The difference between each successive y value
• is a 1,3,5,7,9, pattern
• it can also be a multiple of this

5a
3a
1a
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So lets test the pattern out

• x y -2x²
• 0 0
• 1 -2
• 2 -8
• 3 -18

2
2
6
6
10
10
2 / 2 1 6 / 2 3 So it is a true 10 / 2
5 parabola
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Examine the following graph

• First look at the points

(3,4½)

• Then check the differences

2½

• ½ x 2 1
• 1½ x 2 3
• 2½ x 2 5
• The pattern matches

(2,2)
1½
(1,½)
½
(0,0)
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There is another way to check this!

• x y 2x²
• 0 0
• 1 2
• 2 8
• 3 18

• First
• Difference
• 2
• 6
• 10

• Second
• Difference
• 4
• 4

• If we check the first differences, we get the
  1,3,5,7, pattern.
• but if we check the second differences, they are
  constant!

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Is a Function Quadratic?

• make a table of values
• choose x values 0, 1, 2, 3,
• find the y values
• take the first differences
• take the second differences
• the second differences should all be the same
  number (they are constant)

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Which Equations Form Parabolas?
Yes!
No!

• y x²
• y 2x²
• y -3x²
• y ½x²
• y x² 1
• y x² - 4
• y x² 2x - 1

• y x
• y 2x
• y -x
• y ½x - 2
• y x³ 1
• y x³ - 4
• y vx

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The Parabola Equation Form!

• Each parabola equation has the same form.
• y ax² bx c
• where a, b and c are numbers
• on the next page we will graph an example so that
  you can use it in completing the project where
  you investigate what happens when you change the
  values of a, b, and c

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Graph y 2x² 2x 1

• x y 2x² 2x 1
• -3 2(-3)² 2(-3) 1 13
• -2 2(-2)² 2(-2) 1 5
• -1 2(-1)² 2(-1) 1 1
• 0 2(0)² 2(0) 1 1
• 1 2(1)² 2(1) 1 5
• 2 2(2)² 2(2) 1 13
• 3 2(3)² 2(3) 1 25

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The graph
y 2x² 2x 1
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Your Assignment

• Your first project requires you to graph a number
  of parabolas
• Your goal is to get a sense of what effects there
  are from changing the values of a, b, and c
• When you make the graphs, keep the same scale for
  each one (if a point is off the graph, just
  ignore it).