Quadratics
How could we defend our castle?
Background Info
Name
Quadratic is a Latin term. It is originated from quadratus, the past participle of quadrare which means "to make square."
Who invented the Quadratic Formula ???
Euclid, a Greek Mathematician, created an intellectual concept during 300 BC. Both Pythagoras and Euclid created a well based procedure to find multiple solutions of the quadratic equation.
Basic Quadratic Terminology
A symmetrical open plane cure (a U-Shape).
2. Vertex
The highest point/tip
Format: The X coordinate comes first. The Y coordinate comes second. (X,Y)
3. Axis of Symmetry (AOS)
A line of symmetry for a graph. The axis of symmetry is considered as a mirror to split images on either side.
4. Optimal Value (Maxima/Minima)
The largest and smallest value/point on the parabola. (y=___)
5. X-Intercept
The point which the graph crosses the x-axis. (x,0)
6. Y-Intercept
The point which the graph graph crosses the y-axis. (0,y) Another way of saying X-Intercept is Zeros.
Optimal Value (Maxima/Minima)
X-Intercept (Zeros)
Y-Intercept
Vertex Form
Axis of Symmetry
Vertex Form Equation: y = a (x-h)^2 + k
Vertex: (h,k)
Axis of Symmetry: x = h
Small Note: The Axis of Symmetry is a vertical line, so it has to be written: x = __
Optimal Value (Maxima/Minima)
Vertex Form Equation: y = a(x-h)^2 + k
Vertex: (h,k)
Optimal Value: y = k
Small Note: The Optimal Value is a horizontal line, so it has to be written: y = __
Transformations
1. Vertical Translation
2. Horizontal Translation
3. Vertical Stretch
4. Reflection
X-Intercepts/Zeros & Y-Intercept
To find the X-intercept/ Zeros, you have to understand that the coordinate is (x,0). Since we know this, to find the X-Intercept, the Y value in the equation of the parabola must be 0 to solve for the X value.
y = a (x-h)^2 + k
Y-Intercept
To find the Y-intercept, you have to understand that the coordinate is (0,y). Since we know this, to find the Y-Intercept, the X value in the equation of the parabola must be 0 to solve for the Y value.
Step Pattern
Factored Form
X-Intercept/Zeros
For Instant:
y = (x + 13) y = (x-6)
0 = x + 13 0 = x -6
-13 = x 6 = x
Axis of Symmetry
Formula:
y = a (x-r) (x-s)
Final Formula:
x = r+s divided by 2
For Instant:
y = (x-r) (x-s)
x = (x-6) (x-8)
x = 6 + 8 divided by 2
x = 7
Optimal Value
For Instant:
y = (x-r) (x-s)
x = (x-6) (x-8)
x = 6 + 8 divided by 2
x = 7
Substitute
y = (7-6) (7-8)
y = (1) (-1)
y = -1
Conclusion
The Optimal Value is y = -1. Also the vertex is (7,-1)
Standard Form
Zeroes
Formula:
ax^2 + bx + c
Converted Formula:
b + - square root b^2 - 4(a)(c) divided by 2(a)
For Instant:
Figure out the following X-Intercepts for the Standard Form Equation 3^2 + 12x + 6
Procedures;
1st Step: Substitute the Standard Equation into the Quadratic Formula
-12 +- square root (12)^2 - 4(3)(6) divided by 2(3)
2nd Step: Continue solving it
12 +- square root 144 - 72 divided by 6 // -12 +- square root 72 divided by 6 // -12 +-
8.49 divided by 6 //
Final Step: @ the end you will have two different X-Intercepts
1st X-Intercept:-0.585 // 2nd X-Intercept: -20.49
Axis of Symmetry
-b/2(a)
For Instant:
When finding the AOS for the given Stand Form Equation y = 4x^2 + 12x + 8, you need:
1st: Make the value of B negative (y = 4x^2 + 12x + 8)
Turn 12 into -12
2nd: You must divide the new negative B value by 2(a)
-12 divided by 2(3)
3rd: Finally, you solve everything
-12 divided by 6 = -2
In Conclusion, the AOS of this equation is (-2)
Optimal Value (Maxima/Minima)
For Instant:
Using the value we've received for x, substitute that in and solve.
(4(-2)^2 + 12(-2) + 8)
8^2 - 24 + 8
64 - 24 +8
48
In Conclusion, your Optimal Value is 48.
Completing the square to turn to vertex form
Factoring
Types of Factoring Equations
2. Simple Trinomial
3. Complex Trinomial
4. Perfect Square
5. Differences of Squares