Algebra Period 3 (Tuesday)

Chapter 12-4 Quadratic functions
A QUADRATIC FUNCTION is not y = mx + b

(which is a LINEAR function),
but instead is
y = ax² + bx + c
OR
f(x) = ax² + bx + c
where a, b, and c are all real numbers and
a cannot be equal to zero because
it must have a variable that is squared ( degree of 2)
Quadratics have a squared term, so they have TWO possible solutions also called roots You already saw this in Chapter 6 when you factored the trinomial and used zero products prop.

If the domain is all real numbers, then you will have a PARABOLA which looks like
a smile when the a coefficient is positive or
looks like a frown when the a coefficient is negative.

Graphing quadratics:
You can graph quadratics exactly the way you graphed lines
by plugging in your choice of an x value and using the equation to find your y value.

Because it's a U shape, you should graph 5 points as follows:

Point 1) the vertex - the minimum value of the smile or the maximum value of the frown

The x value of the VERTEX = -b/2a
We get the values for a and b from the actual equation

f(x) = ax² + bx + c

Plug that into the equation and then find the y value of the vertex

Next, draw the AXIS OF SYMMETRY : x = -b/2a
a line through the vertex parallel to the y axis

Point 2) Pick an x value to the right or left of the axis and find its y by plugging into the equation.

Point 3) Graph its mirror image on the other side of the axis of symmetry by counting from axis of symmetry

Points 4 and 5) Repeat point 2 and 3 directions with another point even farther from the vertex

JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END

Parabolas that are functions have domains that are ALL REAL NUMBERS
Their ranges depend on where the vertex is and also if the a coefficient is positive or negative

EXAMPLE: f(x) = -3x² (or y = -3x²)
the a coefficient is negative so it is a frown face
the x value of the vertex (maximum) is -b/2a or 0/2(-3) = 0

the y value of the vertex is 0
So the vertex is (0, 0)
The domain is all real numbers.
The range is y is less than or equal to zero

To graph this function:
1) graph vertex (0, 0)
Draw dotted line x = 0 (actually this is the y axis!)
2) Pick x value to the right of axis of symmetry, say x = 1
Plug it in the equation: y = -3(1) = -3
Plot (1, -3)
3) count steps from axis of symmetry and place another point to the LEFT of axis in same place
(-1, -3)
4) Pick another x value to the right, say x = 2
Plug it in the equation to find y: y = -3(2²) = -12
(2, -12)
5) count steps from axis of symmetry and place another point to the LEFT of axis in same place
(-2, -12)

JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END