3.1 Quadratic Functions and Models
November 06, 2008
3.1 Quadratic Functions and Models
Objectives:
Graph a quadratic function using transformations
Identify the vertex & axis of symmetry of a quadratic function
Graph a quadratic function using its vertex, axis and intercepts Nov 3­1:41 PM
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3.1 Quadratic Functions and Models
November 06, 2008
General form of a quadratic function:
Standard form of a quadratic function:
a ≠ 0
a ≠ 0
Type of graph: Parabola
This form is easier to graph: Standard Form
How to change from general to standard form: Complete the Square
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3.1 Quadratic Functions and Models
November 06, 2008
Example #1: Find the vertex and line of symmetry
Now you can pick out the vertex and axis of symmetry: (Remember the axis of symmetry from the vertex form is
and .)
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3.1 Quadratic Functions and Models
November 06, 2008
General Form:
We can also find the vertex from the general form by using the equation: . Then we substitute this value in for x and solve for y of the vertex. From the general form, remember that is also the equation for the axis of symmetry.
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3.1 Quadratic Functions and Models
November 06, 2008
Standard Form:
What does a, from the standard form, tell us? If a > 0, the parabola opens up and the vertex is the minimum point. If a < 0, the parabola opens down and the vertex becomes the maximum point.
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3.1 Quadratic Functions and Models
November 06, 2008
Example #2: Find the vertex, axis of symmetry, and graph the parabola.
Nov 4­7:38 AM
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3.1 Quadratic Functions and Models
1) Using the general form
November 06, 2008
2) Using the standard form
Vertex:
Vertex:
Axis of Symmetry: y = ­5/4
Axis of Symmetry: y = ­5/4
Nov 4­7:39 AM
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3.1 Quadratic Functions and Models
November 06, 2008
Having the vertex and knowing whether the graph opens up or down is really not enough to accurately graph the parabola. We should also locate the intercepts.
How many are possible? 0, 1, or 2
How can you find these intercepts? Set the quadratic equal to "0" and solve for "x".
Use factoring, quadratic formula, or completing the square to solve for the x­intercepts.
So the x­intercepts are ­1 and Nov 4­10:56 AM
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3.1 Quadratic Functions and Models
November 06, 2008
Graph:
Nov 4­7:40 AM
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3.1 Quadratic Functions and Models
November 06, 2008
You can tell how many x­intercepts there will be by using the discriminant: b2 ­ 4ac
b2 ­ 4ac > 0, there are two real intercepts
b2 ­ 4ac = 0, there is only one (double) real intercept
b2 ­ 4ac < 0, there are no real intercepts
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3.1 Quadratic Functions and Models
November 06, 2008
Example #3:
Graph using vertex, axis of symmetry, y and x­intercepts (if any).
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3.1 Quadratic Functions and Models
November 06, 2008
Example #4: Write the quadratic function with V (3, 0) and containing the point (6, 9).
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3.1 Quadratic Functions and Models
November 06, 2008
Summary of all we have discussed is on page 157.
HW: P.164/22, 29, 31, 33, 45, 49, 51, 55, 57, 59, 63, 65, 67
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