Bellwork

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# Bellwork - PowerPoint PPT Presentation

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
• 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
• Factor
• 2x2+5x+3
• x2+15x-16
Bellwork Solution
• Factor
• x2+15x-16
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=ax2+c

Section 10.1

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
• 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”

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
• 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

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

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

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

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
• 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)

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 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
• 10.1
• 1-5, 7-23 odd, 33-36

Y

X

Example
• Graph
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