Unit 3: Quadratics and other non-linear functions

Solving Quadratic inequalities in one variable

1.) Find the ZEROES of the corresponding quadratic equation... 2.) Plot the zeroes on the number line (x-axis)... 3.) Test the points between and outside the zeroes in the ORIGINAL equation (kind of OPTIONAL)... 4.) Write the solution set

standard form to vertex form

1.) If necessary make the leading coefficient 1... 2.) Find the number that would make a perfect square trinomial. To do this, half b and square it (b/2)^2... 3.) Add and subtract the number from step (2) --> put it in between the second and third terms... 4.) Re-write the trinomial as a binomial squared and combine like terms (and re-isolate "y" if necessary) *Remember: Your leading coefficient must be 1*

Solving absolute value equations

1.) Isolate the absolute value (move other numbers across the = sign)... 2.) Set the "inside" of the absolute value equal to the other side of the equation. Do this again but make one side NEGATIVE (negate the whole side)... 3.) Solve both equations... 4.) Check your solution (plug it back in)

Solving Absolute Value Inequalities

1.) Isolate the absolute value (move other numbers across the inequality sign)... 2.) Solve as is (take away the absolute value and solve the inequality)... 3.) DO this again with the second inequality --> negate one side and flip the sign this time... 4.) Write the solution set and check your solution (You can graph your solutions on a number line and test values - like quadratic inequalities)

The three useful parts of a parabola that can help you sketch its graph:

1.) the Vertex (x = -b/2a, f(-b/2a)) .... 2.) the y-intercept (x=0)... 3.) the x-intercept(s) (y=0)

Axis of Symmetry

A line that divides a plane figure or a graph into two congruent reflected halves. Its a VERTICAL line.

polynomial definition

A monomial or a sum or difference of monomials

inequality

A statement that compares two quantities using <, >, ≤, ≥, or ≠ (When you are graphing an inequality, use a dotted line and a hollow circle for these < > and a solid line and a solid circle for these ≤ ≥).

factored form

An expression expressed as the product of its factors.

Adding and Subtracting Polynomials

Combine *like terms* (same variable and exponent). When subtracting a polynomial, you must subtract each term (distribute negative)

Equation of a Circle:

General form: x^2 + y^2 + Cx + Dy + E = 0... Center-Radius Form: (x - h)² + (y - k)² = r² --> center = (h,k) (h & k have the opposite sign) --> radius = r

The Zero Product Law definition

If the *product* of multiple factors is *equal to zero*, then at least *one of the factors must be equal to zero* (if something equal's zero, then at least one of the something's must be zero) (y=0)

Multiplying Polynomials

Multiply the coefficients, keep the base, and add the exponents (each term of the first polynomial must be multiplied by each term of the second). (ex: x^a times X^b = X^a+b) [Some require double distribution / foil (first, outer, inner, last)--> (2/3x-1)^2 --(subtraction)--> (2/3x-1)(2/3x-1) --(double distribution--> 4/9x^2-4/3x+1]

Special Cases to solving Quadratic inequalities in one variable:

No solution {}, a 0 with a line through it, (ex: x^2 - 6x + 9 <0 --> parabola does not include x-axis) OR All Real Numbers {xIxeR} (ex: -x^2-4<0 --> when graphed the parabola is less than zero everywhere) (parabola is above/below x-axis)

Absolute value must be

POSITIVE, so when it equals a negative, there is no solution.

solution set

The set of all values of the variable that make the equation true / The set of all solutions of an inequality

Roots/zeroes

The solutions of a quadratic equation / A value of x which makes a function f(x) equal 0 (an input value that produces an output of zero). Sometimes the roots are the x-intercept(s) (x,0). They are EQUIDISTANT from the axis of symmetry.

Vertex

The turning point of a parabola (it's a point). (The minimum or maximum) The x-coordinate is always on the axis of symmetry.

Note:

The vertex of the parent function y = x^2 is always (0,0). So, by writing a quadratic in vertex form, we can clearly see how the parabola shifts horizontally and vertically.

interval notation

a way to describe the solution set of an inequality

multiplying polynomials when its a fraction

ex: (3a^4/4b^2)^2 --> exponent goes with both the numerator and denominator (x/y)^3 = (x^3/y^3) --> 9a^8/16b^4

multiplying polynomials power to power

ex: (5x^2yz^3)^2 --> everything gets squared (MULTIPLY exponents) --> (5x^2yz^3)(5x^2yz^3) --> 25x^4y^2z^6

The whole point of solving an inequality is to

find all values of the variable(s) that make the inequality true. (Most of the time, there are an infinite number of solutions to an inequality. Hence, the *solution set* can be written in either *set builder notation* or *interval notation*)

Solving Quadratic inequalities in TWO variables meaning

finding, graphically, all (x,y) coordinate pairs that make a particular inequality true (Shade the graph in the direction based on the inequality symbol - the solution set is the points within a graphs shaded region).

Math conjugate

formed by changing the sign between two terms of a binomial [ex: (a+b) and (a-b), they're conjugates]. The product of conjugates is ALWAYS the differences of the squares (DOTS) of the terms (x+a)(x-a) = x^2-a^2).

if a < 0 (negative - /less than zero), then the function has a

maximum. The parabola opens DOWNWARDS.

If a > 0 (positive +/greater than zero), then the function has a

minimum. The parabola opens UPWARDS.

Finding the zeroes with a graphing calculator:

plug in function/equation into y= on calculator --> graph --> (zoom standard) --> 2nd Trace --> Zero --> left bound of first zero ... enter... right bound of first zero... enter, enter... (repeat for second zero)

Finding the minimum/maximum values of a function with a graphing calculator:

plug in function/equation into y= on calculator --> graph --> (zoom standard) --> 2nd Trace --> minimum/maximum --> left bound of the vertex ... enter... right bound of the vertex... enter, enter...

To find the zeroes/roots of a function

set y = 0 and solve for the x values (since the zeroes are where the function = 0)

The Zero Product Law is used to

solve any factorable quadratic equations. To use: set the equation equal to zero and factor the non-zero side (set y equal to 0)

The absolute value of a number is

the distance that number is away from zero. so if IxI = 7, then x can either be 7 or -7.

set-builder notation

the expression of the solution set of an inequality

Parabola

the graph of a quadratic function

Quadratic Equation word problems

the solution will almost always depend on... 1.) The vertex (min/max)... 2.) The x-intercept(s) (hits the ground)... 3.) The y-intercept (initial height)

to find the axis of symmetry (equation)

use the formula x = -b/2a (this finds the equation of the axis of symmetry which is always x = __ )

To convert a general form to center-radius form, you can

use the method of completing the square - TWICE (once for each variable)... Steps: 1.) Group x terms and y terms. Move constant to the other side (the number without a variable)... 2.) Complete the square on x, then on y (b/2)^2... *Remember... whatever you add to one side you HAVE to add to the other*

How do we find the vertex (point)?

use this (x = -b/2a, f(-b/2a)) (this represents (x,y).... to find the y-value, plug x into the equation)

y-intercept

where a line crosses the y-axis; x=0

x-intercept

where the graph crosses the x-axis; y=0

Vertex Form of a Parabola

y = a(x - h)^2 + k (*Where (h,k) is the vertex --> opposite sign for h/x*)

A quadratic function is any function in the form of

y = ax^2 +bx +c (standard form) where the leading coefficient is NOT zero

you can NEVER factor when

your leading coefficient is NEGATIVE (so, make it positive and switch the other signs ... ex: -2x^2+16x+40 --> 2x^2-16x-40... now factor)

The Zero Product Law is extremely important in finding the

zeroes or x-intercepts of a parabola (remember the x-intercepts occur when y=0)