Math Quadratics

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A (y) COORDINATE

Optimal Value

The Highest(Maximum or Lowest(Minimum) point on the parabola

Parabola meets Axis of Symmetry to split into two equivalent halves

A (x) coordinate

Axis of Symmetry

Direction of Opening

If a<0, the Parabola opens down

If a>0, the Parabola opens up.

QUADRATICS

Differential Values

Table of Values

First Differences

The relationship between the consecutive y-values

A constant relationship states a linear relation

Second Differences

The relationship between the First Difference Values

A constant relationship states a quadratic relation

EXAMPLE

Graphing Parabolas

Plotting Points

A Symmetrical Open Plane Curve

PLOTTING THE VERTEX AND USING STEP PATTERN

Adding 5 more points onto the parabola from your equation

Equations

Y-Intercept

Using 5 Existing Points to find Y-Intercept

Coordiniates of the Parabola meeting the x-axis

0,1 or 2 Roots

Solving of Parabola Characteristics

The (a) value

Vertical Stretch/Compression

A magnitute change to the base parabola

a>1

A Vertical Stretch

0

A Vertical Compression

Vertex

The Roots (x1,x2)

The (h) value with x1+x2/2=h

Subsitituing h into the previous parbola equation to find the y/k of the vertex.

Converting Standard Form to Vertex Form

The (h,k) values

y = a(x-h)^2+k

(h,k) represent V(x,y)

y = 2(x-4)^2+7 V(4,7)

A (y) value

The Optimal Value (k)

A Minimum Value

The parabola opens up and is the lowest point on the parabola.

A Maximum Value

A vertical line that divides the parabola into two congruent halves

The parabola opens down and is the highest point on the parabola.

A (x) value

The Axis of Symmetry (h)

Roots/Zeros

a(x-s)(x-t)=0

setting each factor to zero to solve

2(x-4)(x+5)=0

x+5=0 x=-5

x-4=0 x=4

Discriminant

b^2-4ac

Perfect Square

The Discriminant is a perfect square, thus making it a rational number that does not need the Quadratic Formula.

Nature of Roots

b^2-4ac < 0

There are no Real Roots

Showing Imaginary Relations(i)

b^2-4ac = 0

There is 1 Real Root (Vertex is on x-intercept)

b^2-4ac > 0

There are 2 Real Roots

x=b2-4ac/2a

Solving Quadratic Equations

Finding X Intercepts

Completing the Square

Standard Form to Vertex Form

Isolating X for roots

Factoring

Standard Form to Factored Form

lal>1

Use Complex Trinominal Factoring

a=1

Use Simple Trinomial Factoring

Perfect Square Trinomial

Difference of Squares

a2 – b2 = (a + b)(a – b)

Quadratic Formula

x = -b ± √b^2 - 4ac ________________ 2a

BE

Discriminant is negative, there will be no roots. If 0, 1 root.

Quadratic Equations

Simplied Form

y=x^2

3 Models

Vertex Form

y=a(x-h)^2+k

Standard Form

y=ax^2+bx+c

Factored Form

y= a(x-s)(x-t)