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Bellwork. Factor 2x 2 +5x+3 x 2 +15x-16 Complete the statement A function’s domain is the collection of the _________. A function’s range is the collection of the __________. Bellwork Solution. Factor 2x 2 +5x+3 x 2 +15x-16. Bellwork Solution. Factor x 2 +15x-16. Bellwork Solution.

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bellwork
Bellwork
  • Factor
    • 2x2+5x+3
    • x2+15x-16
  • Complete the statement
    • A function’s domain is the collection of the _________.
    • A function’s range is the collection of the __________.
bellwork solution
Bellwork Solution
  • Factor
    • 2x2+5x+3
    • x2+15x-16
bellwork solution1
Bellwork Solution
  • Factor
    • x2+15x-16
bellwork solution2
Bellwork Solution
  • Complete the statement
    • A function’s domain is the collection of the _________.
    • A function’s range is the collection of the __________.

Inputs or x’s

Outputs or y’s

graph y ax 2 c

Graph y=ax2+c

Section 10.1

the concept
The Concept
  • In chapter 9 we worked with second order polynomials.
  • In chapter 10 we actually graph these functions and use the for analysis
definitions
Definitions
  • A second order polynomial’s graph is what is called a parabola
  • Thus these functions are sometimes called parabolic functions
  • Another term that is used is a quadratic function.
  • They look like this

Interesting fact: Satellite dishes are parabolic in form because of the special properties attributed to these “conic sections”

definitions1
Definitions
  • Much like linear functions, these functions come with their own nomenclature
  • Two important terms
    • Vertex: Highest or lowest point of a quadratic
    • Axis of symmetry: Imaginary line that divides the parabola into two mirrored halves

Axis of symmetry

Vertex

more terminology
More terminology
  • These terms are important because they’re used to describe different parabolas, much like slope was used for lines
  • As well, they are used to describe changes made to the parent function
    • A parent function is a standard graph for a basic function
    • This graph is the parent function for a quadratic
graphing

Y

X

Graphing
  • At this point, graphing these function is best done via T-table

x

y=x2

1

1

2

4

3

9

-1

1

-2

4

-3

9

graphing1

Y

X

Graphing
  • We can also plot several iterations to see the effect of a scalar (or leading coefficient) attached to the term
  • This scalar makes the equation y=ax2

x

y=x2

y=2x2

y=1/2x2

1

1

2

.5

2

4

8

2

3

9

18

4.5

-1

1

2

.5

-2

4

8

2

-3

9

18

4.5

graphing2

Y

X

Graphing
  • These graphs lead us to understand a fundamental of graphing
  • If a>1, the graph stretches
  • If a<1, the graph flattens
graphing3

Y

X

Graphing
  • Let’s look at what happens when a<0

x

y=x2

y=-x2

1

1

-1

2

4

-4

3

9

-9

-1

1

-1

-2

4

-4

-3

9

-9

Therefore we see that if a<0, the graph is mirrored over the x-axis

fundamental rules
Fundamental Rules
  • At this point we see some fundamental rules of quadratics
    • If the leading coefficient is positive (a>0)
      • Concave up (cupped upwards)
    • If the leading coefficient is negative (a<0)
      • Concave down (cupped downwards)
graphing4

Y

X

Graphing
  • Let’s look at one last thing
    • What do you think happens when we add a constant?

x

y=x2

y=x2+2

y=x2-3

1

1

3

-2

2

4

6

1

3

9

11

6

-1

1

3

-2

-2

4

6

1

-3

9

11

6

Therefore we see that the constant dictates the height of the function on the y-axis

fundamental rules1
Fundamental Rules
  • At this point we see some fundamental rules of quadratics
    • If the leading coefficient is positive (a>0)
      • Concave up (cupped upwards)
    • If the leading coefficient is negative (a<0)
      • Concave down (cupped downwards)
    • A constant added indicates the y-coordinate of the vertex
homework
Homework
  • 10.1
    • 1-5, 7-23 odd, 33-36
example

Y

X

Example
  • Graph
most important points
Most Important Points
  • A second order polynomial can be called a quadratic function
  • It’s graph is called a parabola
  • Parabola’s have a vertex and axis of symmetry
  • Leading coefficients either flatten or stretch graphs
  • Negative leading coefficients cause the graph to be mirrored across the x-axis
  • A constant indicates the vertex’s y-coordinate