Quadratic Mind Map

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Quadratic Relations

Quadratic Equations

Solutions

Quadratic formula

X-coordinate of the vertex of a parabola is -b/2a and the equation of the axis of symmetry is x=-b/2a

When x-intercepts and one other point are known

Substitute into the y= a(x-r)(x-s) formula

To find an equation that represents the relation

Write equation in standard form and then factor the left side

Set each factor equal to zero and solve for the unknown

Completing the square

Allows you to find the min or max point of a quadratic relation of the form y= ax^2 + bx + c algebraically

Expressions

Special Products

Sum and the difference of two terms

When you multiply the sum and the difference of two terms, the two middle terms are opposites so they add to zero, so you just multiply the rest.

Perfect squares

When squaring a binomial, you add the two equal middle terms after expansion and square the rest.

Polynomials

Factoring

Perfect Square Trinomial

Factor using this formula: a^2 + 2ab + b^2

Difference of Squares

Factor using this formula: a^2 - b^2 = (a + b) (a - b)

Quadratic expression

ax^2 + bx + c

Find two integers whose product is a * c and whose sum is b. Then break up the middle term and factor by grouping

x^2 + bx + c

First find two integers, whose product of c is the sum to b

Grouping

Factor groups of two terms with a common factor to produce a binomial common factor.

Find the GCF of the Polynomial

First find the GCF of the coefficients and then the GCF of the variable parts

Remove the GCF as the first factor, and then divide each term by the GCF to find the second factor

Model as a binomial product

Expand and Simplify

Use the Distributive Property

Parabola

Properties

Opening

Downward

"a" value is negative

Maximum value

Upward

"a" value is positive

Minimum value

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

How to find

Y-Intercept

Make the x-value as 0

X-Intercept

There are 2 x-intercepts

Can be either positive or negative

Make the y-value as 0

Transformations

"a" value stretch or compress the graph vertically by a factor of a

-1

a>1 or a<-1 = graph is stretched vertically (narrows)

a<0 = reflected in the x-axis

"h" value translated the graph horizontally

h<0 = translated to the left

h>0 = translated to the right

"k" value translates the graph vertically

k<0 = translated downward

k>0 = translated upward

How to draw

Values x may take

Any set of real numbers

Values y may take

If a < 0 then y is less than or equal to k

If a > 0 then y is greater than or equal to k

minimum of 7 points

Use the step method

1a, 3a, 5a

Graph of a quadratic relation, which is U-shaped and symmetrical

Getting Ready for Quadratics

Simplifying Algebraic Expressions

Rewrite the expression

Collect like terms

Distributive Property

Find GCF

Simplifying Radicals

Perfect square GCF

Seperate radicals

Exponent Laws

Negative exponents

Negative reciprocal

Power of a fraction

Power of a porduct

Power outside the bracket

Multiply exponents

Dividing

Subtract exponents

Multiplying

Add exponents

y=x^2

Key Terms

Root (of an equation)

The same as the solution of an equation

The value of the varibale that makes an equation true

Quadratic Equation

An equation in the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a can't equal 0.

Completing the Square

A process for expressing y= ax^2 + bx + c in the form y= a(x-h)^2 + k

Quadratic Expression

A second-degree polynomial

GCF (Greatest Common Factor)

The greatest factor that divides two numbers

Perfect square Trinomial

A trinomial of the form a^2 + 2ab + b^2 or a^2 - 2ab - b^2 that is the result of squaring a binomial

Mirror Point

Point in which the points of one side of the parabola was reflected

Congruent

Equal in size and shape

Dependent Variable

Variable that relies on the other

Independent Variable

Variable that does not rely on the other

Finite Differences

Differences found from the y-values in tables with evenly spaced x-values

Using a table of values chart

Second differences are constant

Vertex

Point on a parabola where the curve changes direction (can either have a max or min value depending on opening)

The (x,y) coordinates of the vertex are the (h,k) values in the vertex equation

Quadratic Relation

A relation whose equation is in the form y=ax^2 + bx + c

Axis of Symmetry

Line that divides a figure into 2 congruent parts

Goes through the vertex

Non-Linear Relationship

A relationship between 2 variables that does not follow a straight line when graphed

Curve of best fit

Smooth curve drawn to approximate the general path or trend in a scatter plot