Algebra Honors (Period 6 & 7)

Quadratic Equations 8-8

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 2 possible solutions (roots)
You already saw this when you factored the trinomial and used zero products property.

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.

What happens as the "a" coefficient gets really big or really small (fraction/decimal)? We'll look at that together on my graphing calculator.

But think, what happened when the "m" (slope) coefficient got big?

The slope got steeper.

So now think that both sides of the U get steeper at the same time.

What's happening to the shape of the U???

Now think, what happened when the "m" (slope) coefficient got tiny?
The slope was a bunny slope.
So now think that both sides of the U are bunny slopes at the same time.
What's happening to the shape of the U???

Putting in standard form:
Standard form is:
y = ax² + bx + c

OR

f(x) = ax² + bx + c

You can't read the sign of a, b, or c until it's in standard form (just like y = mx + b!)

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:

First MAKE SURE THE EQUATION IS IN STANDARD FORM!

y must be isolated on one side and then you can read the a and b coefficients.

y = ax² + bx + c


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


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 IMMEDIATELY to the right or left of the AXIS OF SYMMETRY 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 the axis of symmetry


Points 4 and 5) Repeat point 2 and 3 directions with another point ONE STEP FARTHER from the AXIS OF SYMMETRY.



JOIN YOUR 5 POINTS IN A SMOOTH "U" SHAPE ( not a V shape!)

AND 
EXTEND LINES WITH ARROWS ON END



Parabolas are functions whose domains 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 frowny face


The vertex is called the maximum.


The x value of the vertex is -b/2a 


a = -3 and b = 0 (it's missing!)


The x value of the vertex = -b/2a = -0/2(-3) = 0


Plug the x value of 0 back into the function to find the y value of the
vertex:


y = -3(02) = 0
So the vertex is (0, 0)

The domain is all real numbers.


The range is y is less than or equal to zero (It's a frowny face)



To graph this function:


1) Graph vertex (0, 0)


2) Draw the AXIS OF SYMMETRY –
a dotted line at x = 0 (actually this is the y axis!)


3) Pick x value immediately to the right of axis of symmetry, x = 1


Plug it in the equation to find the y value: y = -3(1) = -3


Plot (1, -3)


4) Count the same 1 step from axis of symmetry on the other side of the axis
and place another point to the LEFT of axis at the same y value
(-1, -3)


5) Pick another x value to the right 2 steps away from the axis of symmetry, x = 2

Plug it in the equation to find y:
y = -3(22) = -12
Plot (2, -12)


6) Count 2 steps from axis of symmetry on the other side of it and
place another point to the LEFT of axis at the same y value
(-2, -12)



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