Quadratic Relations

Key Terms

Non-Linear Relationship

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

Axis of Symmetry

Axis of Symmetry

Line that divides a figure into 2 congruent parts

Goes through the vertex

Quadratic Relation

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

Vertex

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

Finite Differences

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

Using a table of values chart

Using a table of values chart

Second differences are constant

Independent Variable

Variable that does not rely on the other

Dependent Variable

Variable that relies on the other

Congruent

Equal in size and shape

Mirror Point

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

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

GCF (Greatest Common Factor)

The greatest factor that divides two numbers

Quadratic Expression

A second-degree polynomial

A second-degree polynomial

Completing the Square

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

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.

Root (of an equation)

The value of the varibale that makes an equation true

The same as the solution of an equation

y=x^2

y=x^2

Getting Ready for Quadratics

Exponent Laws

Multiplying

Add exponents

Dividing

Subtract exponents

Power outside the bracket

Multiply exponents

Power of a porduct

Multiply exponents

Power of a fraction

Multiply exponents

Negative exponents

Negative reciprocal

Simplifying Radicals

Perfect square GCF

Seperate radicals

Simplifying Algebraic Expressions

Find GCF

Distributive Property

Collect like terms

Rewrite the expression

Parabola

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

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

How to draw

Use the step method

1a, 3a, 5a

minimum of 7 points

Values y may take

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

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

Values x may take

Any set of real numbers

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

Transformations

"k" value translates the graph vertically

k>0 = translated upward

k>0 = translated upward

k<0 = translated downward

k<0 = translated downward

"h" value translated the graph horizontally

h>0 = translated to the right

h>0 = translated to the right

h<0 = translated to the left

h<0 = translated to the left

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

a<0 = reflected in the x-axis

a<0 = reflected in the x-axis

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

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

-1

-1<a<0 or 0<a<1 = graph is compressed vertically (widens)

How to find

X-Intercept

Make the y-value as 0

There are 2 x-intercepts

Can be either positive or negative

Y-Intercept

Make the x-value as 0

Properties

Opening

Upward

Upward

Minimum value

"a" value is positive

Downward

Downward

Maximum value

"a" value is negative

Expressions

Polynomials

Multiplying

Use the Distributive Property

Use the Distributive Property

Expand and Simplify

Model as a binomial product

Factoring

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

Grouping

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

Quadratic expression

x^2 + bx + c

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

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

Difference of Squares

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

Perfect Square Trinomial

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

Special Products

Perfect squares

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

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

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 jus

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.

Quadratic Equations

Solutions

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

Factoring

Write equation in standard form and then factor the left side

Set each factor equal to zero and solve for the unknown

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

Quadratic formula

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