Algebra (Period 1)

Introduction to Quadratic Equations: 13- 1

f(x) = ax² + bx + c where a, b, and c are real numbers and a ≠ 0.
Standard form for a quadratic.
Put 4x² + 7x = 5 in standard form
it becomes
4x² + 7x - 5 = 0

Now you can determine a, b, and c
a = 4
b = 7
c = -6

What about
5x² = -4x

in standard form it is:
5x² + 4x = 0

a = 5
b = 4
c = 0


Example 3.

5x² = -4

5x² + 4 = 0

a = 5
b = 0
c = 4

Solving any
ax² + bx = 0

2x² - 15x = 0
Factor.. think GCF...
5x(4x -3) = 0
ZERO PRODUCTS property
5x = 0 and 4x - 3 = 0
so x = 0 and x = 3/4

{0. 3/4}

Any quadratic in this form a
ax² + bx = 0
has 2 roots
One of the roots is always 0 and the other is -b/a

We tried
10x² - 6x = 0
we found by factoring that we did get x = 0 and x = 3/5


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:
STEP 1: determine 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
just plug in the b and the a value from your equation into -b/2a and you have the x-value of the vertex.
Now to find the y value -- take that x value and PLUG it into the equation


STEP 2: Next, draw the AXIS OF SYMMETRY : x = -b/2a
a line through the vertex parallel to the y axis . Draw this line as a dashed line. REMEMBER: It will be a dashed line parallel to the y-axis


STEP 3: Point 2- Pick an x value to the right or left of the axis and find its y by plugging into the equation.

STEP 4: Point 3- Graph its mirror image on the other side of the axis of symmetry by counting from axis of symmetry

STEP 5: 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)


To graph this function:
1) graph vertex (0, 0)
2) Draw dotted line x = 0 (actually this is the y axis!)
3) 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)
4) count steps from axis of symmetry and place another point to the LEFT of axis in same place
5) pick another x value to the right of the axis of symmetry, say x = 2
plug it in the equation y = -3(2) = -6
plot ( 2, -6). Count the steps from the axis of symmetry and place another point to the LEFT of the axis in the same place.

The domain is all real numbers.
The range is y is less than or equal to zero