Chapter 4,5,6: Quadratic Relations

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Chapter 4: Quadratic Relations

Finite differences

Second differences: The differences between consecutive first differences and so on

First Differences: The difference between consecutive y-values

Quadratic Relations

a relation in the form of y = ax^2 + bx + c a, b and c are real numbers and a can not equal zero

'a' value, 'k' value and 'h' value

The 'a' value can cause a vertical stretch or compression, changing the direction of opening and shape of the curve and/or it can cause a reflection

Right here shows a vertical compression by a factor of 1/2

Reflection a<0

Right here shows a vertical stretch by a factor of 3

The 'k' value controls the vertical translation of the parabola (up and down)

In this case the 'k' value is -1 and the vertical translation was 1 down and horizontal translation was 3 to the right relative to y=x^2

In this case the 'h' value was +4 and a horizontal translation of 4 to the left relative to y=x^2

The 'h' value controls the horizontal translation of the parabola (left and right)

Main topic

Properties of a Parabola

Mirror points

A point that is reflected in the axis of symmetry

Curve of best fit

A curved line to best represent data or a trend in a scatter plot

Line of best fit

A straight line to best represent data or a trend in a scatter plot

Zeroes

They correspond to the x- intercept of the graph of the relation

A value of x which a relation has a value of 0

Maximum value occurs when parabola opens downwards

Minimum value occurs when parabola opens upwards

Vertex, Axis of symmetry, Min/Max value, direction of opening, Values 'x' may take, Values 'y' may take, stretch or compression factor relative to y=x^2, Congruent( = in size and shape)

Some Key definitions

y-intercepts: the y-coordinate of the point where a line or curve crosses the y- axis, at this point x is 0

X- intercepts: The x-coordinates of the point where a line or curve crosses the y- axis, at this point y=0

Linear relations

Straight line

Vertex: A point at which the curve changes direction Congruent: Equal size and shape Parabola: a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side Axis of symmetry: Vertical line drawn through the vertex

Vertical Stretch and Compression

Vertical compression is -1

Vertical stretch is 1

Non Linear relations

y is the dependent variable solved after x affected by the independent variable (x)

x is the independent variable to be solved 1st

Translation

Rotation

A transformation of a geometric shape in which each point is turned about a centre point

Reflection

A transformation in which a point and its image are equidistant from the line of reflection

STEP Method

How to do the step method:

1a, 3a, 5a, and 7a

Relation

May be expressed in ordered pairs, a table of values, a graph, or an equation

An identified pattern between 2 variable

Parabolas

Non-Linear relations

A curved line

Completing the Square

Quadratic Formula.

- If b^2-4ac=0 ! real root or 2 equal roots

- If b^2-4ac<0 no real roots

- If b^2-4ac>0 = to 2 real roots

Used best for precise roots

Solving Quadratic Equations

quadratic formula

always best for precise roots

C.T.S

Can be used always best when in standard form

factoring

Can be used sometimes/ use when C=0

graphing

Can be used always

Roots

also known ass x-intercepts or zeroes

Quadratic Equations

Factored form

What We Can Read: Direction of opening and x-intercepts

Vertex form

What We Can Read: Vertex(h,k), Direction of opening, The min/max value and AOS

Standard form

What can we read: Direction of opening and Y-intercept

3 possible outcomes

Special Products

Greatest common factors are used in factoring polynomial, In fact factoring is the opposite of expanding

Common Factors:

Difference of squares

Perfect square trinomial

Quadratic Expression

Ex: 4x^2+20 and x^2+7x+10 are quadratic expressions

A second degree polynomial

Multiply Polynomials

FOIL: (First, Outside, Inside, Last)

Formula: Y=x^2

Subtopic

Strategies on Solving Word Problems

1.Obtain info from the graph 2. Sketch important parts with whatever information you have 3. Understand

Sketch a graph

Find the vertex, value of 'a', direction of opening

Find The Given.

Simple Quadratics

Variables

Algebra

Distributive Property

Expression

A mathematical phrase made up of numbers and variables, connected by operators ( -, +)

Exponent Law

A set of rules that can be used to simplify powers

Exponent or Power

A raised number to indicate repeated multiplication of a base number