**Mariah Doze's Mind Map**

**Quadratics**

Quadratic Functions and Transformations

Modeling With Quadratic Functions

Identifying Quadratic Data

Factoring Quadratic

Expressions

Quadratic Formulas

Completing the Square

The Quadratic Formula

Square Roots and Radicals

**Quadratic Inequalities**

Standard Form

Intercept Form

Vertex Form

Imaginary and Complex Numbers

Quadratic Systems

Parent Function of a Quadratic Transformation: y=ax^2

Axis of symmetry is x=0

y-intercept is at (0,0)

The vertex form gives you information about

the placement of the parabola on the graph

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

Formula: y=a(x-h)^2+k (a does not equal zero)

Axis of Symmetry: x=x (where x is the x coordinate of the vertex)

Vertex: (h,k)

If a>0 the parabola opens upward; y coordinate of the vertex is the minimum value

If a<0 the parabola open downward: y coordinate of the vertex is the maximum value

y=ax^2 and y-ax^2 are reflections of each other over the x axis

Increasing the absolute value of the a value stretches it vertically and narrows it horizontally

Decreasing the absolute value of the a value compresses it vertically and widens it horizontally

If the k value is positive then the parabola underwent a vertical translation upward

If the K value is negative then the parabola underwent a vertical translation downward

If the h value is positive then the parabola underwent a horizontal shift to the LEFT

If the h value is negative then the parabola underwent a horizontal shift to the RIGHT

Domain is all real number since x is not restricted

Range is all real numbers greater than the minimum value f

or upward facing parabolas or all real numbers less than the

maximum value for downward facing parabolas

Formula: y=ax^2+bx+c (a does not equal zero)

Equations for Finding Vertex:

x=-b/2a

y=f(-b/2a)

(0,c)=y-intercept

You can model quadratic functions algebraically or with a calculator.

Algebraically

With Calculator

Substitute the coordinates of the points into the

model y = ax^2 + bx + c to obtain a system of

three linear equations.

Then use either the substitution or the elimination method to figure out the values of a, b, and c.

Step 1) Press the STAT key.

Step 2) Select EDIT and press ENTER. Enter your data. Usually we put the x-values in L1 and the y-values in L2 .

Step 3) Press STAT key. Select CALC and press ENTER. Select the fourth item which is

LinReg(ax+b),

You can factor quadratic trinomials (ax^2+bx+c) in to products of two binomials.

You can use the distributive property or the FOIL method to multiply two binomials.

When a is 1 you can factor by find factors of the c value that add to get you the b value.

When the a value is not 1 you can factor out the greatest common factor.

Sometimes trinomials will factor out to be perfect square binomials,

which means the binomial ifs the square of a binomial

Sometimes a trinomial will be missing a b value. This usually indicates that the trinomial

is a difference of two squares and the b values were equal and opposite so the canceled

each other out. A tip when factoring these types of trinomials is

a^2-b^2=(a+b)(a-b)

You can also use the AC method to factor. Find factor of the product of

the a value

and the c value that add to get the b value. Then factor

from there, which usually consists of factoring

out the greatest common factor.

Multiplication Property of Square Roots (r=radical):

r(ab)=r(a) x r(b)

Division Property of Square Roots (r=radical):

r(a/b)=r(a)/r(b)

Numbers under the radical have two solutions: one

negative solution and one positive solution.

Quadratic Equations

The x intercepts of a parabola can be called zeros of a function. They are also called the solutions of quadratic equations. They can also be called roots.

The Zero Product Property:

ab=0 then a=0 or b=0

You can find the zero, solutions, roots, or x intercepts of a quadratic equation by putting the equation in to your calculator and viewing it's table or graph, finding the paces where y equals zero.

You can also find the zeros, solutions, roots, or x intercepts of a quadratic equation by hand graphing the equation., finding the paces where y equals zero.

When solving quadratic equations check for false

answers by plugging your answers back

in to the equation.

Completing the perfect square trinomial allows you to factor the completed trinomial as the square of a binomial. You can solve an equation containing a perfect square by finding square roots. The ideal form for solving this type of equation is: ax^2=bx=c

In order to complete the square:

Step 1) Get the equation in the ideal form

Step 2) If the a value is not 1 then factor out the a value

Step 3) Complete the square by adding (b/2)^2 to each side of the equation

Step 4) Factor the perfect square trinomial

Step 5) Find the square roots

Step 6) Re-write as 2 equations

Step 7) Solve for x

Formula: x=(-b +or- r(b^2-4ac))/2

Discriminant: b^2-4ac

discriminant > 0, two real solutions

discriminant < 0, no real solutions

discriminant = 0, one real solution

r=radical

Imaginary unit i is the complex number whose square is -1.

So, i^2=-1 and i is r(-1).

For an positive number a:

r(-a) = r(-1 x a) = r(-1) x r(a) = i r(a)

Imaginary number: Any number of the form a +bi, where a and b are real

numbers and b is not equal to zero.

To graph a complex number locate the real part on the horizontal axis and the

imaginary part on the vertical axis.

The absolute value of a complex number is its distance from the origin and the

complex plane. The absolute value of a+b1 = r(a^2+b^2)

To add or subtract complex numbers combine the real parts and the imaginary parts separately.

Multiply complex numbers as you would binomials.

Complex Conjugate: Numbers pairs of the form

a+bi and a-bi

To divide complex numbers multiply the numerator and denominator by the complex conjugate of the denominator,

then substitute i for -1 or vice-versa.

You can solve quadratic systems algebraically and by graphing

The purpose is to all points where the two graphs intersect.

Use the substitution method to find all points where the quadratic systems intersect.

Check solutions by plugging them back into the original equations.

You can also solve quadratic systems by using a calculator. Simply plug in the equations into your calculator and graph them. Use the INTERSECT feature to see at what points the

the graphs intersect.

Solve Quadratic inequalities algebraically using the substitution

method and then by graphing.

You can also solve quadratic systems in your calculator. Simply plug in to the equations into your calculator making sure to change the equal sign to an inequality sign and graph. The shaded region will be your solution(s).

The greater than or equal to quadratic systems by

The greater than or equal to sign: Sold line

The less than or equal to sign: Solid line

<: Dotted line

>: Dotted line

Shade in the part where the graphs share the same solutions.

Formula: a(x-p)(x-q) (a does not equal zero)

p= x-intercept x value 1

q=x-intercept x value 2

You can find the a value by plugging in the y intercept and the x-intercept into the equation and then solve for the a value.

You can find the y-intercept by plugging in the a value and the x-intercepts into the equation and then solve for the y value (x would be zero since you are solving for the y- intercept).

You can find the axis of symmetry by averaging the x values of the x-intercepts.

Linear vs. Quadratic Data:

For linear data the first differences of adjacent y-values are constant.

For quadratic data the second differences are constant and not equal to zero.

r =radical