TECH 155

Signs, Exponents, and Parenthasis

(-3)^2 = 9 -(3)^2 = -9 -3^2 = -9

Quadratic Word Problems (in Factored Form)

A Rocket launched from a platform, its height (in meters), x seconds after the launch is modeled by: h(x)= -4(x+2)(x-18) What is the height of the rocket at the time of the launch? At the time of the launch is zero seconds, so x= 0: h(0)= -4(0+2)(0-18) h(0)= 144m How many seconds after the launch until the rocket hits the ground? Use the Zero Product Property to find the x-axis: 0= -4(x+2)(x-18) (x+2)= 0 x= -2 (x-18)= 0 x= 18 x= -2 cannot be the solution because the negative implies time prior to launch at x= 0 seconds, therefor, the rocket hit the ground at 18 seconds after launch. How many seconds after launch will the rocket reach its max height? This is asking for the axis of symmetry, to determine, find the average of the x-axis: (-2)+18= 16/2= 8 seconds after launch, the rocket was at its peak height. What is the maximum height the rocket will reach? To determine, find the vertex by plugging in the axis of symmetry into the original equation: h(8)= -4(8+2)(8-18) h(8)= 400m

Quadratic Word Problems (in Vertex Form)

A Rocket launched from a platform, its height (in meters), x seconds after the launch is modeled by: h(x)= -5(x-4)^2 +180 What is the height of the rocket at the time of the launch? At the time of the launch is zero seconds, so x= 0: h(0)= -5(0-4)^2 +180 h(0)= 100m How many seconds after the launch until the rocket hits the ground? Use the Zero Product Property to find the x-axis: 0= -5(x-4)^2 +180 -5(x-4)^2 = -180 (x-4)^2 = -36 x-4 = +/-6 x= +/-6 +4 x= -6+4= -2 x=6+4= 10 x= -2 cannot be the solution because the negative implies time prior to launch at x= 0 seconds, therefore, the rocket hit the ground at 10 seconds after launch. How many seconds after launch will the rocket reach its max height? This is asking for the axis of symmetry, to determine, find the average of the x-axis: (-2)+10= 8/2= 4 seconds after launch, the rocket was at its peak height. What is the maximum height the rocket will reach? To determine, find the vertex by pulling the information from the equation (or plugging in the axis of symmetry into the original equation): h(x)= -5(x-4)^2 +180 (x-4)= 0 x= 4 Vertex: (4,180)

How to go about Factoring

Before starting any factoring problem, it is helpful to write your expression in STANDARD FORM. [Why?] Most factoring skills are introduced using expressions in standard form. Because of this, it may be easier for you to RECOGNIZE factoring patterns when a quadratic expression is in the standard form. Once this is the case, you can proceed to the following list of questions: Q1: Is there a GCF (greatest common factor)? If no, move onto Q2. If yes, factor out the GCF and continue to Q2. (Factoring out the GCF is a very important step in the factoring process, as it makes the numbers smaller. This, in turn, makes it easier to recognize patterns!) Q2: Is there a difference of squares (i.e. x^2 − 16 or 25x^2 - 9)? If a difference of squares pattern occurs, factor using the pattern a^2 - b^2 = (a+b)(a-b). If not, move on to Q3. Q3: Is there a perfect square trinomial (i.e. x^2 − 10x + 25 or 4x^2 + 12x + 9)? If a perfect square trinomial is present, factor using the pattern a^2 ± 2ab + b^2 = (a±b)^2. If not, move on to Q4. Q4: a.) Is there an expression of the form x^2 + bx + c? If no, move on to Q5. If yes, move on to b). b.) Are there factors of c and that sum to b? If yes, then factor using the sum-product pattern. Otherwise, the quadratic expression cannot be factored further. Q5: Are there factors of ac that add up to b? If you've gotten this far, the quadratic expression must be of the form ax^2 + bx + c where a≠1. If there are factors of ac that add up to b, factor using the grouping method. If not, the quadratic expression cannot be factored further.

Rational Exponents

Exponents written as fractions. 4^1/2 =sqrt 4 = 2 9^1/2 = sqrt 9 = 3 8^1/3 = cube root 8 = 2 27^1/3 = cube root 27 = 3

Horizontal Lines

Slope is zero

y-intercept (in equation)

Location where the line intercepts the y-axis. (x,y) (0,5)

Parabola

Shape: Open up (smile) Open down (frown) Vertex is the peak (open down) Vertex is the valley (open up) Axis of symmetry is the folding the graph in half through the vertex.Annotated as x= # Y-intercept (1 point of intersection of the y- axis) X-intercept (0-2 points that are equidistant from the axis of symmetry)

Vertical Lines

Slope is undefined

Slope-Intercept Form to Standard Form

Slope-Intercept Form ==> Standard Form y = mx + b ==> Ax + By = C Multiply by all Denominators: y = 2/3x + 4/7 3(y = 2/3x + 4/7) 3y = 2x + 12/7 7(3y = 2x + 12/7) 21y = 14x + 12 Rearrange into Standard Form: -14x + 21y = 12

Zero Product Property

a*b=0 either a, b, or both a & b must equal 0. (2x-1)(x+4)= 0 either (2x-1) must equal 0 or (x+4) must equal 0 so set each equal to zero and solve: (2x-1)= 0 2x= 1 x= 1/2 (x+4)= 0 x= -4 You are left with two answers: x= 1/2 and -4 To verify, plug in either solution (or both) into the origional equation and solve. you will get 0.

Vertex Form of a Quadratic Function

f(x)= a(x-h)^2 + k where a ≠ 0 and (h, k) are the coordinates of the vertex of the parabola: Smiling Parabola: f(x)= a(x+h)^2 - k Frowning Parabola: f(x)=-a(x-h)^2 + k y= 3(x+2)^2 -27 ***(+ initial term means parabola is shaped up like a cup) First solve (x+2)= 0 x= -2 the x= -2 and the -27 become the vertex: (-2,-27) y= -pi(x-2.8)^2 + 7.1 ***(- initial term means parabola is shaped down like a frown) (x-2.8)= 0 x= 2.8 Vertex: (2.8, 7.1)

Graphing Quadratics in Factored Form

y= 1/2(x-6)(x+2) to determine your X-INTERCEPTS, use the Zero Product Property for all terms with an x: (x-6)= 0 x= 6 (x+2)= 0 x= -2 X-intercepts: (6,0) and (-2,0) To determine the AXIS OF SYMMETRY, find the average of the x-intercepts: 6+(-2)= 4 4/2= 2 Axis of symmetry: x= 2 To determine the VERTEX, plug the axis of symmetry into the original equation: x= 2 y= 1/2(x-6)(x+2) y= 1/2(2-6)(2+2) y= -8 Vertex: (2,-8)

Decimal Notation (standard notation)

the normal way of writing numbers. 3.979*10^4 = 39790 1.2*10^-2 = 0.012

Evaluating Fractional Exponents

(25/9)^1/2 25^1/2 / 9^1/2 sqrt25/sqrt9 5/3

Evaluating Quotiant of Fractional Exponents

256^4/7 / 2^4/7 (256/2)^4/7 (128)^4/7 (7root 128)^4 2^4 16

Multiplying Binomials by Trinomials

(10a - 3)(5a^2 + 7a - 1) Multiply one term at a time from the binomial, WATCH SIGNS!: 10a(5a^2 + 7a - 1) -3(5a^2 + 7a - 1) Distribute: 50a^3 + 70a^2 -10a -15a^2 -21a + 3 Combine termines with like degrees: 50a^3 + 55a^2 -31a +3

Cube Roots

The opposite of the cube of a number. Finding the non-negative number, when cubed, equals the number under the radical sign.

The Slope of a Line (steepness of the line or rate that the line is increasing or decreasing)

rise/run or delta y/delta x rise = change in y (y2-y1) run = change in x (x2-x1)

Compound Inequalities: AND

3x+7 < 2x AND 4x+8 > -48 x < -7 AND x>-14 The portion of the number line that is shaded-in will overlap between -7 AND -14, with no other section of the number line shaded in. An AND compound inequality requires both equations be satisfied. No solution occurs when the shaded-in portions of the number line do not overlap. This means no single number can satisfy BOTH of the original equations: 5x-3 < 12 AND 4x+1 > 24 x < 3 AND x > 6

The Parts of a Polynomial Expression

3x^2 -8x +7 terms: 3x^2, -8x, 7 coefficients: 3, -8, 7 exponents: 2, 1, 0

Compound Inequalities: OR

5x+7 < 27 OR -3x < 18 x < 4 OR x > -6 The portion of the number that is line shaded-in will overlap between 4 and -6, but the entire number line will satisfy either one OR both of the initial equations. SELECT THE ANSWERS THAT OVERLAP! 3x-91 > -87 OR 21x+17 > 25 x > 4/3 OR x > 2 solvig for x and choosing 1 answer: x > 4/3

Function

A relationship from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. The action or actions that an item is designed to perform. input ==> FUNCTION ==> output A function is anotated by the format of: f(x) FUNICTIONS can have strict : f(x) = 2x if the value is odd, or x+5 if the value is even input: x= 1 f(x) = 2(x) f(1) = 2(1) = 2 (output aka the solution) input: x= 2 f(x) = x+5 f(2) = 2+5 = 7 (output) A FUNCTION CAN ONLY HAVE ONE OUTPUT FOR A GIVEN INPUT!!!

Absolute Extrenum Points on a Graph

Absolute max and min points on a graph can be the start/finish of the function's depicted domain interval. If the max or min has a straight line, they are all the absolute Extreneum pointsas long as the line is horizontal. if it is vertical, it is not a function. (Multiple y values plotted in correspondence to a single x value.

Real Numbers

All rational and irrational numbers. (pi, sqrt(2), 1/2, -0.25)

Simplifying Square Root Expressions

First, simplify if possible: 2sqrt(7x) * 3sqrt(14x^2) Combine like terms (factor at the same time): 2*3* sqrt(7*2*7*x*x*x) Simplify again if possible: 2*3*7*x sqrt(2*x) 42xsqrt(2x) If the root is a cube root or higher, you would need that number of numbers to get it out of the root. 5root(96) 5root(2*2*2*2*2*3) 2*5root(3)

Scaling and Reflecting Absolute Value Functions: Equations

if you have the graph of y= |x| slope is 1, therefore, at x=2 y=2. If the question states that it is reflected across the x-axis, it means now y= -2. The new equation would be y= -|x| and is drawn like an upside down V with the peak at the origin with a slope of -1. If the question also states that the equation is scaled vertically by say a factor of 7, it means 7 times the y-value at a given x. The new equation of the graph would be y= -7|x|. The graph would become a very thin upside down V whose peak is at (0,0) with a slope of -7.

Multiplying Monomial Challenge

(3x^a)(bx^4)= -24x^6 Isolate appropriate terms from either side of the equal sign: 3b= -24 and (x^a)(x^4)= x^6 Solve: b= -8 and x^a+4=6 ==> a= 2 Verify by plugging in answers: (3x^2)(-8x^4)= -24x^6 -24x^6= -24x^6

Squaring Binomials

(a+b)^2 is NOT a^2 = b^2 (a+b)^2 IS (a+b)(a+B) and FOIL method DO NOT distribute the square: WRONG: (x + 7)^2 => x^2 + 7^2 => x^2 + 49 Correct: (x + 7)(x + 7) => x^2 +7x +7x +49 => x^2 +14x +7

Evaluating Mixed Radicals and Exponents

6^1/2 * (5root6)^3 6^1/2 * 6^3/5 6^1/2+3/5 6^11/10

Area of a Circle

A = πr² r = d/2 d=2r C = 2rpi = dpi c/d = pi

Linear Equations

The exponent on each variable is 1. often used in a conversion formula. Helpful in graphing linear functions for data trend observations.

Writing Exponential Functions

The format is g(t)= A*r^t A = the Initial Value r = the Common value Write the function fot g(t) with the inital value of -2 and the common value of 1/7: g(x)= -2*1.7^t

The Slope of a Line

The rate of change of one variable to another. m= y1-y2/x1-x2 Positive slopes increase and negative slopes decrease (looking left to right).

A table with an x column and a y column

Cannot have repeating values. Therefore, this table cannot be representative of a function. it has a repeating x value, there it is proven to not be a function. x | y James | 5'11 Nathan | 6'1 Kenneth | 6'2 Peter | 5'10 Dean | 6'1 Kenneth | 5'11

Convert a Decimal to a Percent

Multiply by 100 and add a percent sign. 0.01*100 = 1% 0.241*100 = 24.1% 0.4*100 = 40%

Exponential Function Word Problem

There 170 deer on a reservation whose population increases by 30% per year. Write a function: P(0)= 170 (initial value) P(1)= 170+30%*170 = 170+ 0.3*170 = 170 (1+0.3) = 170 (1.3) P(t)= 170 *(1.3)^t P(2)= 170 *(1.3)^2 P(3)= 170 *(1.3)^3

Integers

WHOLE numbers and their opposites ie: -3, -2, -1, 0, 1, 2, 3

Percent Problems

is/of = percent/100 proportion: A/X = B/100% A= B*X/100%

Vertical Lines DO NOT represent a Function.

x--> f(x) ---> y one x input through a FUNCTION can only output one y A vertical line represents one x having an infinite amount of y's having a relationship with it, which defies the key attribute of a function: having one output for every input.

Exponential Function Initial Value and Common Ratio

If you put zero in for the exponent: f(x)= 5*3^x f(x)= 5*3^0 f(x)= 5*1 Initial Value= 5 The common ratio is found by: f(x)/f(x+1)= 5*3^x+1/5*3^x = 3 the base of the (domain) exponent If given as: Initial Value: 5 Common Ratio: 6 f(x)= 5*6^x

Multiplying Monomilas by Polynomials Challenge

-2y(y^2 + cy - 3) = dy^3 + 12y^2 + fy Distribute: -2y^3 + -2cy^2 + 6y = dy^3 + 12y^2 + fy Isolate appropriate terms by matching their degrees on either side of the equal sign: -2y^3 = dy^3 solve for d: d= -2 -2cy^2 = 12y^2 solve for c: c= -6 -6y = fy solve for f: f= 6

Two-Variable Quadratics

1st check if there are any common factors. 2nd Rearrange to simplify Grouping step 3rd Perform grouping, remember to look at terms like (4y)x the same as (4)x REMEMBER: Slow is Fast, and Fast is Slow

Convert a Fraction to a Percentage

Divide out fraction, multiply by 100 and add a percent sign. 23/10 = 2.3 2.3*100 = 230% 2/5 = 0.4 0.4*100 = 40%

Area of a Square, Rectangle, and Parallelogram

base multiplied by height A = bh

Undefined Domains

A domain will be rendered undefined if it results in the function having a fraction whose denominator is 0 or a sqrt with a negative number. f(x)= 1/sqrt(6-|x|) If the value of x is greater than 6, the sqrt will be negative and therefore the function will be undefined. If the value of x is equal to 6 the denominator will be zero and therefore the function again is undefined. Domain: -6 < x < 6 *(if no absolute value had been included in the function, then the Domain would have been 0<x<6)

Intercepts of a Line

A point where a line intersects a coordinate axis is called an intercept. The x-intercept is the point on the line at which y=0. The y-intercept is the point on the line ate which x=0. The find x-intercepts, plug 0 in for y. To find y-intercepts, plug 0 in for x. Determine the x-intercept by plugging in y=0: 2y + 3x = 6 2(0) + 3x = 6 x=2 therefore the x-intercept is (2,0) Determine the y-intercept by plugging in x=0: 2y + 3x = 6 2y + 3(0) = 6 y=3 therefore the y-intercept ia (0,3)

A Function

A rule that takes an input (domain) and assigns a unique output. f(x)=3x+7 x would be a real number (except 0)

Meaning of Terms in Word Problems

Addition: x+3 plus, sum, more than, increased by Subtraction: x-3 minus, difference, subtracted by, less than, decreased by Multiplication: 3x times, product, of, by Division: x/3 divided by, quotient, per Solutions: x=3 x>3 x<3 equals, is, evaluate, solve for, simplify

Inequalities

Algebraic statements that have ≠, <, >, ≤, or ≥ as their symbols of comparison. When dividing or multiplying by a negative number, the inequality symbol must switch: -12x < 24 x > -2 When adding or subtracting a negative number, the inequality symbol remains the same: x-12 < 24 x < 36 On a number line, open circles represent less-than and greater-than inequalities; while colored in circles represent less-than-or-equal-to and greater-than-or-equal-to inequalities.

Relative Extremum Points of a Graph

All peaks (maximum) and valleys (minimum). WATCH OUT make sure they are actual peaks not just sharp edges of a slope.

Rational numbers

All positive and negative integers, fractions and decimal numbers. (sqrt(2), 1/2, -0.25)

DUMMY CHECK for FUNCTIONS!!!

Always insert 0 into the function to ensure that it does NOT come out undefined. If it does turn out to be undefined when x= 0 is plugged in, then the answer MUST include x=/= 0! Watch out for absolute values in the function!! test out your answer so as to ensure your interval is actually correct!!

What is the Range of a Function?

The Range of a Function f(x) is the set {y=f(x)}, aka all possible outputs (y = solutions). If f(x)= x^2 DOMAIN: all Real Numbers RANGE: f(x)> or equal to 0 If g(x)= x^2/x = x but DOMAIN must include x is any real number, but cannot equal (=/=) 0. RANGE: All real numbers, but f(x)=/= 0.

Exponential Growth and Decay Word Problems

DECAY: Suppose a radioactive substance decays at 3.5% per hour. What percent of the substance is left after 6 hours? First, make a table: hours | Percent remaining 0 | 100% 100-3.5= 96.5% = 0.965 (lost per hr) 1 | 0.965(100) amount lost after 1st hr 2 | 0.965*0.965(100) lost 2nd hr = 0965^2(100) ....n | 0965^n(100) n=6 0.965^6(100) = 80.75% left after 6 hours GROWTH: Jessica owns a chain of cupcake shops that operated 200 stores in 2018. If the rate of increase is 8% annually, how many shops will Jessica own in 2026? years | Percent increase (2018) 0 | (100%) 200 (2019) 1 | (100% + 8%= 108% = 1.08) 1.08(200) (2020) 2 | 1.08*1.08(200) = 233.28 ....(2026) 8 | 1.08^8(200) = 370 shops owned in 2026 The DIFFERENCE is you would take the initial y and subtract the initial x for a decay problem but add for a growth problem.

Dividing Scientific Notation (coefficient must be between 1 and 10!!!)

Divide the coefficients and subtract the exponents. (7*10^5)/(5*10^7) divide the coefficients: 7/5 = 1.4 divide the exponents: 10^5/10^7 = 10^5-7 = 10^-2 multiply the coefficient by the exponent: 1.4*10^-2

The Function f(x) is grahed what is it's domain and range?

Domain: look at the interval displayed regarding the X-AXIS and answer in the form of: -2 <or equal to x < 3. Range: look at the interval displayed regarding the Y-AXIS and answer in the form of: -1 < f(x) < or equal to 7. less than or equal to again id depicted by filled-in circles and < is depicted as open circles.

Compund Inequalities: Double

Double is another version of the AND compound inequality. -16 < 3x+5 < 20 can be rewritten to: (THIS IS EASIER TO PLOT!!!) -16 < 3x+5 AND 3x+5 < 20 -7 < x AND x < 5 Double compound inequalities may also be solved by maintaining the x between the two inequality symbols: -16 < 3x+5 <20 -7 < x < 5

Factoring Quadratics: Difference of (Perfect) Squares

Every polynomial that is a difference of squares can be factored by applying the following formula: a^2 − b^2 = (a+b)(a−b) Note that 'a' and 'b' in the pattern can be any algebraic expression. For example, for a=x and b=2 we get the following: x^2 − 2^2 = (x+2)(x−2) ​ The polynomial x^2 − 4 is now expressed in the factored form, (x+2)(x−2). We can expand the right-hand side of this equation to justify the factorization: (x+2)(x−2) ​= x(x−2)+2(x−2) = (x^2 − 2x) + (2x − 4) = x^2 − 4​ The leading coefficient does not have to equal to 1 in order to use the difference of squares pattern. In fact, the difference of squares pattern can be used here! This is because 4x^2 and 9 are perfect squares, since 4x^2 = (2x)^2 and 9 = (3)^2. We can use this information to factor the polynomial using the difference of squares pattern: 4x^2 − 9 = (2x)^2 − (3)^2 = (2x+3)(2x−3) *** if it is not a perfect square, check to see if factoring out the GCF makes it into the "perfect" square required to use the Factoring Quadratics: Difference of (Perfect) Squares method.

Multiplying Binomials using the FOIL Method

F first O outside I inside L last (3x + 2)(5x - 7) First 3x * 5x = 15x^2 Outside 3x * -7 = -21x Inside 2 * 5x = 10x Last 2 * -7 = -14 Combine terms with like degrees and put into standard Polynoial form (Ax^2 + Bx - C): 15x^2 - 11x - 14

Simplifying Cube Root Expressions

First Simplify if possible and convert cube roots into fractional exponents: 5* 3root(2x^2) * 3* 3root(4x^4) 5*3 * (2x^2)^1/3 * (4x^4)^1/3 15* (2x^2 *4x^4)^1/3 15* (8x^6)^1/3 15* 8^1/3 *x^2 15*2*x^2 30x^2

The Average Rate of Change Problems

Given a GRAPH and asking which interval the average rate of change equals -4 with the answer in the interval form of #<x<#: Option 1: -1<x<1 Find the (x,y) for x= -1 and x= 1. Solve for slope by (y2-y1/x2-x1). the average rate of change for (-1,7) and (1,-1) is -4 Given a TABLE and asking what the average rate of change is for the interval #<x<#: For Example, the interval -5<x<-2: Find the points for x= -5 and x= -2. Determine the slope of those points (-5,6) and (-2,0) have a slope of -2, therefore, the average rate of change is -2. Given a EQUATION and asking which interval (in the form of #<x<#) has the average rate of change of 1/2: For Example, the equation y=1/8x^3-x^2 Option 1:-2<x<2 Plug x=-2 and x=2 into the equation and make and x|y table with the results. find the slope from those two points (-2,-5) and (2,-3). The average rate of change is 1/2. Given a WORD PROBLEM and Tables. Solve the same as a table and compare solutions to find the answer to question asked by the word problm.

Writing Linear & Exponential Functions from 2 Points

Given: (-1,9) (1,1) Linear Function: f(x)= mx+b *Find m (change of y over change of x): 1-9= -8 1--1= 2 -8/2= -4 *Plug one of the points in to find b: 9= (-4)(-1) + b b= 5 f(x) = -4x+5 Exponential Function: g(x)= Ar^x *One at a time, plug in each set of points: g(-1)= Ar^-1 = 9 A/r= 9 or A= 9r g(1)= Ar^1 =1 Ar= 1 *Plug in A= 9r into Ar= 1: (9r)(r)= 1 9r^2= 1 r^2= 1/9 sqrt(r^2)= sqrt(1/9) r= 1/3 *Plug r=1/3 into A= 9r A= (9)(1/3) A= 9/3 A=3 g(x)= 3*1/3^x

Shifting Absolute Value Graphs

If shifting to the right, x must minus that amount. If shifting to the left, x must plus that amount. For instance, if you have the equation y= |x| graphed, and you are asked to shift the graph to the right 3, the new equation becomes y= |x-3| and the x-intercept shifts from (0,0) to (3,0). If shifting up in the verticle direction, your original equation would go from y= |x| to y= |x| +#. For instance, if you were asked to shift the equation up 4 and over 3, the new equation would become y= |x-3|+4 If given y= |x| and were asked to shift it down 4 and to the left 3, the new equation would be y= |x+3|-4.

Rules of Exponents

If this gets too confusing, rewrite it without the exponent "^": x^m + x^n = x^(m+n) x^m * y^m = (xy)^m (x^m)^n = x^nm x^0 = 1 x^1 = x x^-m = 1/x^m (x^m)/(x^n) = x^(m-n) (x^m)/(y^m) = (x/y)^m

Domain of a Radical Function

The domain must be a real positive number, meaning x must be greater than or equal to zero (whichever has the sq root sign) f(x) = sqrt(2x-8) Domain: 2x-8>/= 0 2x>/= 8 Domain of f(x)= sqrt(2x-8) is: x>/= 4

Determining the Y-intercept from a given table instead of an equation

Lines have a constant pattern. when looking at a given table, observe the pattern: x|y -2|8 -1|? (answer is 6) 0|? (answer is 4) 1|2 2|0 3|-2 4|-4 moving vertically down the x-side of the table, you notice that it increases by 1, signifying that the change of x = 1. Moving vertically down the y-side f the table, you notice that it decreases by 2, signifying that the change of y = -2. thus determining that the y-intercept is (0,4)

Polynomial

Means a sum of many terms. RULES: These terms CANNOT have negative, fractional, or non-number exponents, and no square roots i.e. 10x^-7 -9x^1/2+ 15x^a+ sqrt4 +9 (only the +9 is an acceptable term) The "COEFFICIENT" is the number in front of the variable in the given term: 9x^2, 9 is the coefficient. (REMEMBER, if it is a number by itself like 9x^2 +7, both 9 and 7 are coefficients. 7 could be considered 7x^0 which is the same as 7*1 (because of x^0=1). Number of approved terms | i.e. monomial- one term | pib^5, 6x^0, 10z^15 binomial- two terms | 9a^2-5, 3y^3-5y trinomial- three terms | 7y^2-3y+pi The "DEGREE" of the term is determined by the power of the exponent of each term: for a monomial like 10x^7 the degree of this term is 7. when you are given a binomial or larger, to find the degree of the polynomial, you look at the term with the highest value exponent: for the binomial 10x^8 + 9x^2, the degree is 8 The "LEADING COEFFICIENT" the first term's coefficient when the polynomial is written in STANDARD FORM aka degree order starting with the term with the greatest degree first: 10x^7 + 9y^6 - 8z^5 + 7, the leading coefficient is 10.

Review of Factorization Methods

Method: Factoring out common factors. Example: 6x^2 + 3x = 3x(2x+1). When is it applicable?: If each term in the polynomial shares a common factor. Method: The sum-product pattern Example: x^2 + 7x + 12 = (x+3)(x+4) When is it applicable?: If the polynomial is of the form x2+bx+cx^2+bx+cx2+bx+cx, and there are factors of c and that add up to b. Method: The grouping method Example: 2x^2 + 7x + 3 = 2x^2 + 6x + 1x + 3 = 2x(x+3) +1(x+3) = (x+3)(2x+1) When is it applicable?: If the polynomial is of the form ax2+bx+cax^2+bx+cax2+bx+ca, x, start superscript, 2, end superscript, plus, b, x, plus, c and there are factors of acacaca, c that add up to bbbb. Method: Perfect square trinomials Example: x^2 + 10x + 25 = (x+5)^2 When is it applicable?: If the first and last terms are perfect squares and the middle term is twice the product of their square roots. Method: Difference of squares Example: x2−9=(x−3)(x+3) When is it applicable?: If the expression represents a difference of squares.

Multiplying with Fractions

Multiply straight across! Numerators times numerators and denominators times denominators including any signs: 5/9 * 3/15 * -2/3 numerators: 5*3*-2= -30 denominators: 9*15*3= 405 -30/405 simplifies to -2/27

Multiplying Scientific Notation (coefficient must be between 1 and 10!!!)

Multiply the coefficients and add the exponents. (9.1*10^6)(3.2*10^-5) ==> 9.1 * 10^6 * 3.2 * 10^-5 multiply the coefficients: 9.1 * 3.2 = 29.12 multiply the exponents: 10^6*10^-5 => 10^6+-5 = 10^1 multiply the coefficient by the exponent: 29.12*10^1 = 2.912*10^2

Positive or Negative Slopes

Negative Slope: (opposite signs b/w top & bottom) delta y = a negative (positive) number over delta x = a positive (negative) number Positive Slope: (same signs for top & bottom) delta y = a positive (negative) number over delta x a positive (negative) number from left to right, is the line extending up (positive) or down (negative)?

Intervals and Interval Notations

Number lines are the visual representations of an interval. (A set of numbers.) It can either be an open, closed or mixed interval. An open interval is written like: -3<x<2 and : (-3, 2) On the number line or graph, open circles will represent that the -3 and 2 are not included in the interval. A closed interval is written like: -3<(or equal to) x, (or equal to) 2 and: [-3, 2] On the number line or graph, filled-in circles will represent that the -3 and 2 are included in the interval. A mixed interval will have both and is written like: -3< (or equal to) x < 2 and: [-3, 2) On the number line or graph, a filled-in circle will represent the -3 is included in the interval, and an open circle will represent that the 2 is not included in the interval. If all numbers but one are included in an interval, it would be written like [-infinity, 1) or (1, + infinity] and the 1 would be drawn with an open circle on the number line or graph and all other values shaded in.

Irrational Numbers

Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating. (like pi)

Point-Slope Form to Standard Form

Point-Intercept Form ==> Standard Form y-y1 = m(x-x1) ==> Ax + By = C Distribute: y-3 = 3/4(x-5) y-3 = 3/4x - 15/4 Multiply by all Denominators: 4(y-3 = 3/4x - 15/4) 4y - 12 = 3x -15 Rearrange into Standard Form: -3x + 4y = -3

What is the Difference between an Equation and a Function?

Previously we have been looking at equations written as y= x+1. This is a function. it is telling us: y= f(x) = x+1 What is plugged in the x value is written into the left column of the table and the output is written into the right column of the table: x | y f(x) = x+1 => y= ? 0 | 1 f(0) = 0+1 => y= 1 1 | 2 f(1) = 1+1 => y= 2 2 | 3 f(2) = 2+1 => y= 3

Negative Exponents

Raising a number to a negative exponent is the same as raising the number's reciprocal to the equivalent positive exponent. 3^-1 = 1/3 2^-2 = 1/4 -27^-1/3 = -1/3

Convert Percentage to Decimal

Remove the percent sign (%) and divide by 100. 40% 40/100 = 0.4

Methods of Factoring Polynomials

Showing how term "factor" is used in several different processes: Factor monomials by writing them as a product of other monomials: 12x^2= (4x)(3x) Factor the GCF (the Greatest Common Factor) from polynomials using the distributive property: 2x2+12x=2x(x+6) Factor out common binomial factors which resulted in an expression equal to the product of two binomials: x(x+1) + 2(x+1) = (x+1)(x+2) Writing the polynomial as a product of 2+ factors. So in all three examples, we indeed factored the polynomial.

What Each Linear Equation is Best Used For

Slope-Intercept Form- identifying the y-intercept and the slope immediately. Point-Slope Form- identifying the slope immediately. Standard Form- the simplest way to solve for x-intercept and y-intercept. I find the slope-intercept form the easiest to work with, but it all depends on what information is given, and what the question is asking for. all forms can be rearranged and solved into the slope-intercept form if desired.

What is the Domain of a Function and makes it undefinable?

The set of all possible input values for a function. if you have: f(x) = 2/x and the input or Domain is x=0 f(0) = 2/0 is an undefined answer The output to the function cannot be undefined, therefore x cannot equal zero. Certain contingencies can be placed on a function: h(x) is 1 if x=3 or h(x) is 2 if x=6 therefore the only possible answers to this function are: h(3) = 1 OR h(6) = 2 otherwise written as "domain: {3, 6}" annotating that these two numbers are the only valid inputs to this specific function. If specific contingencies do not allow for x to equal say 4, h(4) is undefined. Any numbers besides x= 1 and x=2 are undefined for this function.

Scientific Notation Rules (coefficient must be between 1 and 10!!!)

The shorthand method of writing really large or really small numbers. If the decimal is moved to the right, the exponent gets smaller. If the exponent is POSITIVE, the decimal moves to the RIGHT. 34750000000= 3.457*10^10 1*10^1 = 10 1*10^2 = 100 3.102*10^2 = 310.2 120,000 = 1.2*10^5 If the decimal is moved to the left, the exponent is gets larger. If the exponent is NEGATIVE, the decimal moves to the LEFT. 0.000000003457 = 3.457*10^-10 1*10^-1 = 0.1 1*10^-2 = 0.01 3.102*10^-2 = 0.03102 .000012 = 1.2*10^-5

Factoring Quadratics: Perfect Squares

To expand any binomial, we can apply one of the following patterns: (a+b)^2 = a^2 + 2ab + b^2 (a-b)^2 = a^2 - 2ab + b^2 The reverse of this expansion process is a form of factoring. If we rewrite the equations in the reverse order, we will have patterns for factoring polynomials of the form a^2 ±2ab + b^2. a^2 + 2ab + b^2 = (a+b)^2 a^2 - 2ab + b^2 = (a-b)^2

A human body has 5 liters of blood, 40% of which is red blood cells. Each red blood cell has a volume of approx. 90*10^-15 liters. How many RBCs are in the human body? (Round to 2 decimal places answering in scientific notation.) (coefficient must be between 1 and 10!!!)

Total blood = 5 li RBCs = (40%)(5li) = 2li Volume of each RBC = 90*10^-15 li (90*10^-15 in scientific notation= 9.0*10^-14.) 2li/9.0*10^-14li = 0.222*10^14 converted to scientific notation = 2.22*10^13

Linear Word Poblems

Using the function equation, determine to points from the word problem, solve for b (the y-intercept (0, b) and m (the slope delta y/delta x): F(t)= mt + b A lake is covered by an ice layer 2 meters thick in winter. As the ice melts in spring, the layer of ice decreases at a constant rate. (RATE IS A KEYWORD MEANING SLOPE!) After 3 weeks the layer is only 1.25 meters thick. S(t) denotes the ice layer's thickness, so S(t)= mt + b at the beginning of spring (t= 0), the ice thickness was 2 meters ==> b= 2. S(0)= m(0) + 2 meters From this, we identify one point as (0, 2) Also given is at three weeks the thickness is 1.25 meters, from this information we can pull out the second point needed (3, 1.25) Once two points are pulled from the word problem, using delta y/ delta x solve for m. (0, 2) delta y: 1.25 - 2/ -0.75/3 (3, 1.25) delta x: 3- 0 m= -0.25 Rewrite the equation as: S(t)= -0.25t + 2

Intro to Factors and Divisibility

What are the factors of 12? (aka what whole numbers DIVISIBLE by 12 equal whole numbers?): 1*12, 2*6, and 3*4 all = 12 => 1, 2, 3, 4, 6, and 12 are all factors of 12. 12/12= 1 12/6= 2 12/4 =3 12/3= 4 12/2= 6 12/1= 12 ALL WHOLE NUMBERS

Standard Form of a Linear Equation

When X and Y are variables and A, B, and C are integers: Ax + By = C The standard form is most useful in finding the x-intercept. by making a table with y=0 and x=0, substituting them into the equation one at a time and solving, you can easily find the x and y-intercepts. These intercepts are two points on the graph, and the slope can also be found from them: 9x + 12y = 72 9x + 12y = 72 9(0) + 12y = 72 9x + 12(0) = 72 12y = 72 9x = 72 y-int= (0,6) x-int= (8, 0) slope = delta y / delta x: 0-6/8-0 = -6/8 m = -3/4

Radical Expressions and how to Rationalize the Denominator

When you have a fraction with a radical in the denominator, you can use a technique called rationalizing a denominator to eliminate the radical. Multiply numerator and denominator by the radical that converts the denominator into a square root of a perfect square. -6/sqrt7 -6*sqrt7/ sqrt 7*sqrt 7 = -6sqrt7/7

Piecewise Functions

a function composed of 2 or more equations in which different equations are used for different intervals of the domain. intervals expressed as: f(x) { y, #<x<# or y, (x,x]} it is drawn using open circles(greater /less than) or closed circled (greater/less than or equal) and a horizontal line. f(x) {-9, (-9,-5]} -9, 6, and -7 are the range and { 6, (-5,-1]} (-9,-5],(-5, -1], and (-1,9] the domain {-7, (-1, 9]} anything not covered by the domain is considered undefined. If an equation is given instead of a value in the y position, it will not be a horizontal line b/w the interval. f(x) {-0.125x+4.75 {-10,2)} solve -0.125(-10)+4.75= 6, y=6 -0.125(2)+4.75 = 5, y=5 a line for the interval has a closed circle at point (-10,6) and an open circle at point (2,5)

Exponential Function

a function that can be described by an equation of the form y= a*b^x, where b > 0 and b ≠ 1: the input of the function (the domain) is in the exponent: y=3^x x | y -4 | 3^-4 1/3^4 1/81 -3 | 3^-3 1/3^3 1/27 -2 | 1/9 0 | 1 1 | 3 2 | 9 3 | 27 4 | 81 The negative x values get smaller and smaller but never actually touch zero and the positive x values cause a rapid increase in y values. Exponential functions do not have the linear slope that makes a linear function a straight line on a graph, instead, it has sharp curves.

The Average Rate of change

aka SLOPE of secant line between two points, use to estimate instantanous rate of change at a point. Change of distance/ change of time is a rate. SLOPE IS A RATE. The slope of the two end-points of an interval is the net slope (aka the average rate of change). f(b)-f(a)/b-a from a<x<b (aka [a,b]

Linear Equation: y=mx+b

an equation between two variables that gives a straight line when plotted on a graph. y=2x+3 x|y 0|3 1|5 2|7 plot the (x,y) pairs on a graph with an x-axis and y-axis, a straight line will form. All sopossible solutions to a linear equation will appear on the line that 2 or more points form.

Radical Expressions

an expression that contains a square root: sqrt x*sqrt y is the same as sqrt x*y sqrt x/sqrt y is the same as sqrt x/y (y cannot = 0)

Intro to Grouping (Distributive Property in Reverse)

ax^2 + (a+b)x + b 6x^2 + 7x + 1 First, find the a*b and a+b: a * b= 6 * 1= 6 a + b= 7 Second, find the factors of 6 that when added together, make 7: a | b 1 | 6 1 + 6= 7 Third, "GROUPING" group the terms on the side that has a common factor with it: (ax^2 + ax) + (bx + b) (6x^2 + 6x) + (1x + 1) Fourth, factor out common binomial factors: 6x(x+1) + 1(x+1) (6x+1)(x+1) Finally, verify the factorization is correct by redistributing and matching to original Polynomial expression given: (6x+1)(x+1) 6x^2 + 6x + x + 1 6x^2 + 7x + 1

Dummy Check for Grouping Method

ax^2 + (a+b)x + b: +a & +b ax^2 - (a+b)x + b -a & -b ax^2 - (a+b)x - b -/+a & +/-b ax^2 + (a+b)x - b -/+a & +/-b

Increasing, Decreasing, Positive, or Negative Intervals

f(x) is INCREASING when y is increasing with x f(x) is DECREASING when y is decreasing with x still increasing f(x) is POSITIVE when x>0 f(x) is NEGATIVE when x<0

Dividing with Fractions

flip the second fraction and multiply straight across! 5/9 divided by 3/15: 5/9 * 15/3 numerators: 5*15= 75 denominators: 9*3= 27 75/27 simplifies to 25/9

Finding the Function's Interval for a Piecewise Function

g(x){ x+7 -6<x</=-3 } { 1-x -3<x<4 } { 2x-1 4</=x</=6 } To determine the DOMAIN, verify all real numbers in the first and final expressions include all the rest.: -6<x</= 6 To determine the FUNCTION's interval for each provided domain and range set, put each x into the range: HOWEVER, YOU MUST WATCH THE PLACEMENT OF THE X IN THE RANGE! if the x is before the operations sign, as is with the first set provided, you would plug in the domains minimum value into the range to know the function's minimum value and the domains maximum value into the range to determine the functions maximum value: g(x) { x+7 -6<x</= -3 } MIN DOMAIN: (-6)+7 = 1 MAX DOMAIN: (-3)+7 = 4 1<g(x)</=4 if x is after the operation sign, as with the second set provided, you would plug in the domains maximum value into the range to determine the functions minimum value and the domains minimum value into the range to determine the function's maximum value: g(x) { 1-x -3<x<4 } MAX DOMAIN: 1-(4) = -3 MIN DOMAIN: 1-(-3) = 4 -3<g(x)<4 the third set provided's range has the x before the operational sign same as the first: g(x) { 2x-1 4</=x</=6 } MIN DOMAIN: 2(4)-11 = -3 MAX DOMAIN: 2(6)-11 = 1 -3</+g(x)</=1

Writing an Exponential Function from a Table that is not Consecutive

h(n)= A*r^n n | h(n) 2 | 144 4 | 324 6 | 729 To find the Common Value (r), use ratios: First Ratio: h(4)/h(2) 324/144= 9/4 Second Ratio: A*r^4/A*r^2= (the A's cancel out) = r^2 Put first and second ratios equal to each other: 9/4= r^2 sqrt(9/4)= sqrt(r^2) r= 3/2 To find the Initial Value (A), add to the given table: n | h(n) 0 | ? =64 1 | ? = 96 2 | 144 4 | 324 6 | 729 Because we know the Common value we can find the next to numbers for h(n). (When n= 0, h(n) is the initial value, A): To find h(1)= ? we divide h(2) the common value h(1)= 144/(3/2)= 96 h(0)= 96/(3/2)= 64 OR To find the Initial Value (A), plug in h(2)'s values and solve for A: h(2)= A*(3/2)^2= 144 A*9/4=144 A= 64 h(n)= 64*(3/2)^n

Absolute Value Graphs

is a v-shaped graph that shows that if x=3, then both x=3 and x=-3 will have the same y value. y= |x| has a slope of 1 y= -a|x-b|+c a (Keyword: SCALED)tells us to stretch the original y=|x| graph by a factor of a (aka it is the slope). The negative in front dictates the reflection of the original graph over the x-axis (from a V-shaped graph to an upside-down V-shaped graph.) (If the a was positive, the graph would remain in a V-shape, keyword: REFLECTED). b (keyword: Shifted)tells us to shift the original graph to the right by b spots. (If it had been +b the graph would slide to the left) c (keyword: Shifted)tells us to shift the original graph up c spots. (if it had been -c the graph would slide down c spots).

Rewriting Roots as Rational Exponents

sqrt v^6 can be rewritten as v^6 *^1/2 or v^6/2 or V^3 cube root v^2 can be rewritten as v^2 *^1/3 or v^2/3 7root v^3 can be rewritten as v^3 *^1/7 or v^3/7 6root g^5 can be rewritten as g^5 *^1/6 or g^5/6 1/ 8root g^2 can be rewritten as g^2 8^-1/8 or g^-2/8 or g^-1/4

PEMDAS "Order of Operations"

the correct order in which to solve an equation: working left to right repeatedly: Always start with Parentheses, next any Exponents; then Multiplication OR Division; then Addition OR Subtraction. solve inside the parenthesis first following the rules of PEMDAS: (3*1)^2 + (15-6/2)^2 - 7 (3)^2 + (15-3)^2 - 7 (3)^2 + (12)^2 - 7 eliminate any Exponents next: (3)^2 + (12)^2 - 7 9 + 144 - 7 Complete the orders of operations from left to right: 9 + 144 - 7 153 - 7 146

Area of a triangle (all types: right, obtuse, and acute)

triangles duplicated and set hypotenuse to hypotenuse create either a square, rectangle, or parallelogram A =1/2bh

Rules of Inequalities

when divide or multiply by a negative number, SWITCH the INEQUALITY SIGN! <> solve for x: x<3y-2x x-2x<y -x<3y x>-3y

Slope-Intercept Form of a Linear Equation

y = mx + b (m is the slope and b is the y-intercept) so you can draw a line starting from the y-intercept (aka "b") using "m". m=delta y / delta x (rise/run): So for y = 2x + 2 the slope is 2/1 so from the y-intercept at (0,2) you would go UP 2 and to the right 1 for a second point to draw a line on your graph. For y = -1/3x - 3 the slope is -1/3 so from the y-intercept at (0,-3) you would go DOWN 1 and to the right 3 for a second point to draw a line on your graph. If the given equation is not initially in proper slope-intercept form, simply rearrange it to the y = mx + b form: 6x + 3y = 9 becomes y = 2x + 3 y - 3 = 2(x-1) becomes y = 2x + 1 8x = -4y - 16 becomes y = -2x + 4 If "m" is (+), then the slope is positive, meaning from left to right the line move upwards. If "m" is (-), then the slope is negative, meaning from left to right the line moves downward.

Point-Slope Form of a Linear Equation

y-y₁ = m(x-x₁) where m is the slope and (x₁, y₁) is a given point on the line. Give the point-slope equation for m = 2 and (3,2): y-y1 = m(x-x1) y-2 = 2(x+3) The point-slope form can easily be distributed and solved into slope-intercept form: y-2 = 2(x+3) y-2 = 2x + 6 y = 2x + 8

Exponential Growth vs Exponential Decay

y=Ar^x |r| > 1 means the graph will depict growth |r| < 1 means the graph will depict decay Exponential growth: f(x)= 3(2)^x Exponential Decay: f(x)= 3(1/2)^x