Properties of Graphs of Quadratic FunctionsParabola: the curved graph of a quadratic function
Vertex: the point on a parabola where a minimum or maximum y-value occurs.
Axis of symmetry: a line in which a parabola is reflected onto itself.
Vertical stretch: a ratio that compares the change in y-values of a quadratic function with the corresponding y-values of y=x2
Quadratics can be expressed in different forms:
1. Transformational
2. Standard
3. General
Transformational form:
21hxky
a
a = vertical stretch
k = vertical translation
h = horizontal translation
Standard form: khxay 2)(
General form: cbxaxy 2
Review Squaring Binomials and Factoring
2
2
2
2
2
2
1
3
5
2
3
x
x
x
x
x
Factor:
2510
12
3612
168
2510
2
2
2
2
2
xx
xx
xx
xx
xx
Finding the Maximum and Minimum Value The vertex gives you the maximum or
minimum value. Putting quadratics in transformational form
makes finding the vertex easy
Creating the Transformational form of a Quadratic:Completing the Square
•Divide all terms by ‘a’
•Move ‘c’ to the other side
•Add half of ‘b’ squared to both sides.
•Factor both sides
382 xxy
484 2 xxy
Determining Quadratic Functions from Parabolas
If the vertex and at least one other point of a parabola are known, the transformational form of the quadratic function can be found.
−3 −2 −1 1 2 3 4 5
−3
−2
−1
1
2
3
4
5
6
x
y
Roots of Quadratic Equations
Finding the roots of a quadratic means solving the equation.
Roots, zeros, solutions The value of x that makes the equation
equal to zero.
Method 1: Graphing
Let equation equal zero Use TI-Calculator Enter equation into y= CALC:zeros TABLE
Method 2: Factoring by Decomposition
01625 2 xx
Method 3: Completing the Square
01625 2 xx
Quadratic Formula
There is another way to determine the roots that will always work.
Quadratic Formula:
It is used when the quadratic is in general form:
a
acbbx
2
42
0,2 acbxaxy
Imaginary numbers:
What is the square root of -4???
Can’t find the square root of a negative number, so the answer is imaginary.
A complex number is made up of a real number and an imaginary number: a+bi
Some quadratics have no real roots. Therefore the roots are imaginary.
The Number of Roots of a Quadratic Equation The expression b2-4ac in the quadratic formula
is called the discriminant. The discriminant is used to determine the type of
roots a quadratic will have. If the discriminant is larger than zero, the
quadratic has 2 distinct real roots. If the discriminant is zero, the quadratic has one
root, or two equal real roots If the discriminant is less than zero, the
quadratic has imaginary roots.
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