Quadratic Functions and its Families
Homework: pg 192 # 1-3, 4ac, 5bc, 6, 8, (10), 11
Quadratic Functions and its Families
Quadratic functions are often expressed in one of three ways; standard form, vertex form, and factored form.
From each form, we can extract unique details about the graph of the corresponding parabola.
|Standard form |Vertex Form |Factored Form |
|y = ax2 + bx + c |y = a(x - h)2 + k |y = a(x – x1)(x – x2) |
|a ( direction of opening and step pattern |a ( direction of opening and step pattern |a ( direction of opening and step pattern|
|c ( y-intercept |(h, k) ( vertex |x1 and x2 are the |
| | |x-intercepts |
A quadratic function can be changed to the different forms using algebraic techniques;
• Standard Form ( expand and simplify
• Vertex Form ( completing the square
• Factored Form ( factoring
To determine the family of quadratic functions for given criteria means to create a general equation that allows for some variation to account for all possible cases.
Example 1
Determine the family of quadratic functions that have x-intercepts
of –2 and 5.
Example 2
Determine the family of quadratic functions that have a vertex
of (4, 8).
Example 3
Determine the specific quadratic function that has x-intercepts of 1 and 5 and passes through the point (3, 12).
Example 4
Determine the equation of the parabola that has a vertex at (5, -1) and passes through the point (6, 3).
Example 5
Express the following quadratic function in all three forms.
Fill in the information below and graph the function.
[pic]
|Direction of opening | |
|Step pattern | |
|Maximum/Minimum | |
|x-intercept(s) | |
|y-intercept | |
|Vertex | |
-----------------------
y
x
................
................
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