# INTRO TO QUADRATICS

## Solving Quadratic Equations

## By Factoring

## By Completeing the Square

Example:

-5x²+20x-2=0

1. Block off the first two terms→ (-5x²+20x)-2=0

2. Factor out the A→ -5(x²-4x)-2=0

3. Add "zero" by: dividing the middle term by 2, and squaring it→ -5(x²-4x+4-4)-2=0

4. Bring out the negative→-5(x²-4x+4)+20-2=0

5. Factor inside the brackets and simplify→-5(x-2)²+18=0

6. Divide equation by the number in front of the brackets to get rid of it→(x-2)²-18/5=0

7. Bring the second term over to the other side, and square root all to get rid of the square on the first term→√(x-2)²=√18/5

8. Isolate x, and simplify→x=±3.6+2

9. You will get two answers because the number that was square rooted is positive and negative**→ **x=5.6 or x=-1.6

## By Using the Quadratic Formula

## Properties of Quadratic Relations

## Important Terms and Characteristics

__Parabola__→the graph of a quadratic relation

__Vertex__→the point on the graph with the greatest y-coordinate (if the graph opens down) or the least y-coordinate (if the graph opens up); vertex is directly above or below the midpoint of the x-intercepts (if the parabola crosses the x-axis)

__Optimum Value__→the y-coordinate of the vertex corresponds to the optimum value, either a maximum or minimum value

__Minimum Value__→ y-coordinate of the vertex when the graph opens up

__Maximum Value__→ y-coordinate of the vertex when the graph opens down

__Direction of Opening__→ if the constant value of the second differences is positive, the parabola opens up; and if its is negative the parabola opens down

__Axis of Symmetry__→ a vertical line that passes through the vertex (perpendicular bisector) that makes the parabola symmetrical

__Equation of the Axis of Symmetry__→ As the coordinates of the vertex are (h,k), the equation would be x=h since the line passes through the vertex

__Zeros/x-intercepts__→ the x-coordinates of the points the parabola crosses on the x-axis

## Graphing Factorable Quadratic Equations

## Three Things to Consider When Graphing

2. y-intercept: set x=0, and determine this point

3. vertex: the maximum or minimum point of the parabola; very important when solving optimization questions

The video below will demonstrate how to find these three things when your equation is factored, and use them to graph your quadratic equation.

## Stretching, Reflecting and Translations of Quadratics

## Parent/Original/Base Parabola

## Stretching

__Vertical Stretch__

- parabola gets thinner

- y-values increase

- if the A value is greater than 1

__Vertical Compression__

- parabola becomes wider

- y-values decrease

- if the A value is less than 0 and greater than 1 or less than -1 and less than 0 (fractions)

## Reflecting

## Translations

In y=x²+k:

- the "k" produces a vertical translation up or down

- the parabola does not stretch or compress (as there is no "a" value)

- affects the y-values

In y=(x-h)²:

- the "h" horizontally translates the parabola left or right

- affects the x-values

## Different Forms of Quadratic Equations

## Standard Form

A value:

-if the parabola opens up or down

-vertical stretch or compression

C value:

-y-intercept

## Factored Form

A value:

-opens up or down

-vertical stretch or compression

R and A values:

-x-intercepts

## Vertex Form

y=a(x-h)²+k

A value:

-opens up or down

-vertical stretch or compression

H value:

-x-coordinate of vertex

-horizontal shift

-axis of symmetry

K value:

- y-coordinate of vertex

- vertical shift

- optimum value