类别 全部 - factoring - polynomials - solutions - expressions

作者:Patel Sheil 7 年以前

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Quadratic Mind Map

Quadratic relations and equations are fundamental topics in algebra, focusing on finding solutions and understanding the properties of parabolas. The x-coordinate of a parabola's vertex is given by the formula -b/

Quadratic Mind Map

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