Accuplacer Math Terms Accuplacer Math Terms

Accuplacer Math Terms Accuplacer Math Terms Accuplacer Math Terms Define - "origin" = (0,0) = the point where two axes meet Explain - "quadrants" = when the two axes (x and y) cross they form 4 quadrants Explain what - "the distance" means Define - "distance formula" for a line(s) = "distance" - the length between two points on a plane - (x1, y1), (x2, y2) * does not have to be forming a line *distance is ALWAYS positive = (Distance (d) = "square root" all over - [( x2-x1)^2 + (y2-y1)^2] Explain what - "the midpoint" means Define - "midpoint formula" for a line(s) = "midpoint" - the middle point between two points on a plane - (x1, y1), (x2, y2) -that are forming one line *remember - "MidPOINT" = Midpoint (M) = [(x1+x2)/2] , [(y1+y2)/2] Coordinate Plane = ______________ demential. = 2D Describe/Define what a "one demential (1D) line" is In a 1D line, what two things need to be accounted for and explain how we can do this = a horizontal or vertical line = (x=0) - horizontal - (look at the location of (x) on the coordinate plane) = (y=0) - vertical - (look at the location of (y) on the coordinate plane) = Distance and Midpoint: = horizontal = (D = ("absolute value" X2-x1)) = (M = (x1+x2)/2) = vertical = (D = ("absolute value" y2-y1)) = (M = (y1+y2)/2) Define/Explain - "slope" of a line = (m)=(rise/run)=(change in y/change in x) = (y2-y1/x2-1) When looking at a graphed line - how can we immediately know if the slope is negative or positive = (-) "negative slope" = (line starts in Q3 - the line ends in Q1) = (+) "positive slope" = (lines starts in Q1 - the line ends in Q4) Horizontal Lines have ____________ slope. = (0) Vertical Lines have ____________ slope. = "undefined", "no slope", "infinite" Give the - Standard Form of a Linear Equation Describe how to quickly find the - "slope" and "y - intercept" of this specific equation = (Ax + By = C) *rule: A or B can NOT = (0) *rule: A can NOT be "negative" (-) = "slope" = (-A/B) = "y-intercept" = (C/B) Give the - Standard Form of a Quadratic Equation = (ax^2 + bx + c = 0) Give the - General Form of a line - and describe what it essential means = (Ax + By +C = 0) *rule: A can NOT = (0) = "some expression" [= (0)] - expression ALWAYS equals 0 Describe what the "Standard Form of a Polynomial means" = equation is ordered from (largest - smallest) according to the coefficients degrees Define what the "x-intercept" is = point of the (x) axis = (y=0) Define what the "y-intercept" is = point of the (y) axis = (x=0) Define the - Point-Slope Form of a line What is usually given in the problem that triggers to use this specific form = given = slope (m) and a point (x1, y1) *remember: the form of the line is based off what is given - "point-slope" = [m = (y - y1) / (x - x1)] Define the - Slope-Intercept Form of a line What is usually given in the problem that triggers to use this specific form = given = slope (m) and the y-intercept (b) *remember: the form of the line is based off what is given - "slope-intercept" = (y=mx+b) What kind of a line is defined as - (y=mx) = a line that passes through the origin Define - the "slope" of 2 - Parallel Line & - the "slope" of 2 - Perpendicular Line = 2 parallel line's slopes = SAME = 2 perpendictular line's slopes = NEGATIVE RECIPROCAL : [(1/x) - (x/-1)] How would you - "solve by graphing" these two lines: (2x+y=8) and (2x+3y=12) = graph by using - "intercepts" = find the (x) & (y) intercept for the first line: - set (x) in the first equation to (x=0) - then solve for (y), so now you have (0,y) - which is the "y-intercept" - set (y) in the first equation to (y=0) - then solve for (x), so now you have (x,0) - which is the "x-intercept" - take the points (x,0) and (0,y) and graph the first line - repeat the steps above for the second line equation = now that you have two lines graphed you can find the - "point at which they meet " = [(P) = (x,y)] - ANSWER - you can check your answer by plugging it into the two given equations Define - (h,k) = (h,k) = the "center" point of a circle - a "fixed point" Define what the - distance from (h,k) to any (x,y) point is = (r) = radius State the - "Radius (r) Formula" = Radius (r) = "square root all over" : [(x-h)^2 (y-k)^2] According to the - "Equation of a Circle" - define (x,y) = (x, y) = any point on the circumference of a circle Give the - "Equation of a Circle" = [(r^2) = (x-h)^2 + (y-k)^2] Define - "Straight Angle" = 180 (degrees) - (singular) Define - "Reflex Angle" = MORE - than 180 (degrees) - but LESS - than 360 (degrees) - (singular) Define - "Supplementary Angles" = angles that "add" up to = 180 (degrees) - (plural) Define - "Complementary Angles" = angles that "add" up to = 90 (degrees) - (plural) Define what the term - "congruent angles" means = angles that are the SAME in - (radians and degrees) - they DO NOT have to be going in the same direction - they DO NOT have to be on similar sized lines If - line (t) - crossed through 2 "parallel lines": What is formed from this? Define the similarities Define the term used for - (t) = (8) angles are formed = 2 angles that are - "side by side" - add up to 180 (degrees) = 2 angles that are - "diagonal" or "across from each other" are EQUAL to - each other AND those in same positioned angles in the 2nd group = (t) = "transversal" Define the term for - 2 angles that are - "diagonal" or "across from each other" = "vertical angles" When given a Triangle define the - "capital letters" (A,B,C) and "lower-case letters" (a,b,c) What is the trend between the "capital letters" (A,B,C) and "lower-case letters" (a,b,c) = "capital letters" (A,B,C) = ANGLES = "lower-case letters" (a,b,c) = SIDES = the same letters - ex:(A and a) - are across from each other on a triangle Area of a Triangle 1/2bh Triangles adds up to = 180 (degrees) Define - Median = a line drawn on a triangle - starting: from an angle - ending: in the middle (midpoint) of the angle's opposite side - ex: starts at angle A, ends in the middle of side a Define - Scalene Triangle = angles and sides all UNEQUAL Define - Isosceles Triangle = 2 equal sides - the 2 sides being the "LEGS" of the triangle and the "BASE" being UNEQUAL In a Triangle: - define the term of the - "peak"/the highest angle - what are the other 2 angles called - what must be true about the - largest (#) angle and largest (#) side - what is the important rule about the sides of a triangle = "vertex angle" = "base angles" = they must be across from each other - ex:(A and a) = (c) is always the "hypotenuse" and the largest (#) side - but sides (a) and (b) must be greater than (c) when added together Define - Pythagorean Theorem = (a^2 + b^2 = c^2) List the 2 - Pythagorean Triple "groups" and explain the trick to finding all the pythagorean triple triangles = 3-4-5 = multiply the entire group by (1-5) to find the 5 PT's from the group = 5-12-13 = multiply the entire group by (1 and 2) to find the 2 PT's from the group List the 5 other PT's that are not in either group = 8-15-17 = 7-24-25 =20-21-29 =9-40-41 =11-60-61 Define the 2 - "Special Right Triangles" and state the facts about each ... ALL ABOUT TRIG: STUDY, STUDY, STUDY!!!! 1 degree = ____________ radians = (pi) / 180 1 radian = ______________ degrees = 57 degrees Define - "radian": (by - "if you laid it on the circumference of a circle") = if you laid "one radian" on the circumference of a circle and angle would be formed which is defined as - "one radian" What units are involved in trig functions = radians (pi) = degrees How can be "convert": - from (radians) ----> (degrees): - from (degrees) ----> (radians): "degrees" = multiply by - (pi)/180 = "radians" "radians" = multiply by - 180/(pi) = "degrees" 7 common - "conversions/equivalents" of (degrees & radians) 1. (30 degrees) = (pi)/6 2. (45 degrees) = (pi)/4 3. (60 degrees) = (pi)/3 4. (90 degrees) = (pi)/2 5. (180 degrees) = (pi) 6. (270 degrees) = 3(pi)/2 7. (360 degrees) = 2(pi) What is ALWAYS true about (r) = it is ALWAYS "positive" (+) When drawing a trig function on a coordinate plane (x, y) what should we ALWAYS do first = draw the base of the triangle along the (x) axis) Define the - (Greek letter - "theta") Where does it start & end? = an angle formed = ("theta") - ALWAYS starts on the - positive (+) x-axis and travels in the "counterclockwise" direction = ("theta") - can end & form an angle anywhere in (Q1, Q2, Q3, or Q4) as long as it started on (+x) and moves "counterclockwise" Define (r) in a "coordinate plane" - Define (r) in a "right triangle" of a trig function - Define the "formula" of finding (r) according to trig functions - (r) = the "distance" between the origin (center) of a coordinate plane (x and y-axis) to - any pt. (x,y) - in a - "right triangle" - (r) = the side of the hypotenuse (c) = radius (r) = "square root over all" (x^2 + y^2) Draw a 4 triangles in a coordinate plane - one in each quadrant - and define all of the variables/factors ... In a right triangle problem - (lets say we were given two sides) - how do we find the value of the third side = pythagorean theorem - (a^2 + b^2 = c^2) Define the - 2 "special right triangles" - and all of their key variables/factors = (30-60-90) degrees: - which ever acute angle = (30 degrees) - side adjacent to it = "the shortest leg of the triangle" = (1) - the "hypotenuse" = (double the smallest side) = (2) - the "longer leg" = (square root 3) * the (30 degree angle and the 60 degree angle) can be on either acute angle - AS LONG AS - its corrispinding (x) and (y) values accurately agree with the (x and y-axis) = (45-45-90) degrees: - (BOTH - 45 degree angles) are adjacent from - sides that = (1) - (an easier way to say it): = the legs = (1) and the hypo. = (square root 2) Define the - (6) Basic Trig Names & Ratios - (x, y, and r) and include "theta": *(hint!) - easiest way to answer = to draw a triangle in a coordinate plane - and than define trig ratios 1. (sin "theta") = (y/r) 2. (cos "theta) = (x/r) 3. (tan "theta") = (y/x) 4. (csc "theta") = (r/y) 5. (sec "theta") = (r/x) 6. (cot "theta") = (x/y) If given a triangle is drawn a the coordinate plane - and we are given the values of - triangle sides: (r) and (y) - how do you find side (x) Once you have (x) then what do you also have = use formula: - (x) = ("+, -") "square root all over" (r^2 - y^2) - to find (x) just use the (r) formula but isolate (x) to the left side = ALL - 6 Basic Trig Functions How do you - "rationalize the denominator" = multiply BOTH - the denominator and numerator - by the denominator - then the denominator cancels out and you are left with the value of the multiplied numerator Define all the multiples of 90 When - angle "theta" - is a multiple of 90 (degrees) what is (r) = (90, 180, 270, 360) - (r) = 1 - once you have (r) - you define (x) and (y) according to their location (+/-) and then you can define the 6 basic trig functions Sketch: (y = sinx) - refer to - page (230) When should you - "rationalize the denominator" - when taking the accuplacer = EVERY TIME - when there is a "root" or a (-) number - in the denominator Sketch: (y = cosx) - refer to - page (230) sin90 (degrees) = = 1 cos90 (degrees) = = 0 csc90 (degrees) = = 1 sin(pi)/2 (radians) = = 1 cos(pi)/2 (radians) = = 0 tan90 (degrees) = = ("undefined", none or infinity) - because the denominator = 0 csc(pi)/2 (radians) = = 1 cos0 (degrees) = = 1 sec0 (degrees) = = 1 sin0 (degrees) = = 0 csc0 (degrees) = = 0 tan0 (degrees) = = 0 cot 0 (degrees) = = 0 cos180 (degrees) = = (-1) sec180 (degrees) = = (-1) cos(pi) (radians) = = (-1) sec (pi) (radians) = = (-1) Define - "periodic" in terms of trig functions = ALL trig functions are "periodic" - meaning they when graphed trig functions are represented by a curve that goes in one full "period" (aka completion) - then it does it again What is the "period" for: (sin, cos, and tan) & (csc, sec, cot) - (sin, cos) & ( csc, sec) - period = 360 (degrees) or 2(pi) (radians) - (tan, cot) - period = 180 (degrees) or (pi) (radians) When graphing/sketching - (sin & cos) - what formula should you use immediately - [(y) = Asin (Bx + C) + D] - [(y) = Acos (Bx + C) + D] - "absolute value" [A] = the "amplitude" or "height" of the curves in the graph - "period" = (360/B) - in degrees, or (2(pi)/B) - in radians - this solves for how many times the graph repeated its cycle/curve Describe whats different about the graph of - (y=Asinx) - when (A>0): vs. (A<0): - with "SIN" - (y=Asinx) if: - (A>0) "positive" - than the curve will start like normal (0,0) and first head upwards - (A>0) "negative" - the the curve will still start like normal (0,0) but instead it will head downwards **depending if (A) is positive or negative will also effect where the curve will stop - (but thats not as important) Describe whats different about the graph of - (y=Acosx) - when (A>0): vs. (A<0): - if (A>0) "positive" than the curve is - "right side up" - if (A<0) "negative" than the curve is - "upside down" **(note!: the curve that is actually "right side up" looks "upside down) and vice versa -so make sure to not get confused!) In equation - [(y) = A(sin/cos) (Bx + C) + D] - how do you solve for a "left-right shift": = (-C/B) - it will always be in parenthesis when given [(y) = Asin (Bx + C) + D] - define - "absolute value" [A]: Explain how to find it using its own formula: - "absolute value" [A] = the "amplitude" or "height" of the curves in the graph - so if: [A] = (6) and was in the formula - (y=6cosx) - that would mean that the highest point of the curve is at (6) and the lowest is at (-6) - [A] = (max - min)/2 In equation - [(y) = A(sin/cos) (Bx + C) + D] - how do can we recognize an "up-down shift": - "up-down shift" = (D) - we can recognize it because it will always be added or subtracted after the parenthesis - if (D) is "positive" - the graph is moving "UP" - but if (D) is "negative" - the graph is moving "DOWN" Before a "function" - can be defined as a "function" - what MUST happen = the functions's graph must pass the "vertical line test" - meaning: you can draw a vertical line anywhere on the graph, if that line hits the graph in more than one spot, then the graph is NOT a function - if NOT = it has passed & is a function According to sketching trig functions - (sin & cos) - explain what these two graphs would look like: 1. (y=10sin4x) : - define the (4): 2. (y=(-7)sin(x/5)) : - define (x/5): = in - (y=10sin4x) - the (4) before the (x) means that the graph is reaching 360 (degrees) 4X faster - so we would need to solve for the "period" = (360/4) = 90 (degrees) = its "period" - in - (y=(-7)sin(x/5)) - the (x) is being divided by the (5) - which means that the graph is "stretched out" 5X more than it would be in (y=sinx) - so we would need to solve for its "period" - [2(pi)/(1/5) = 10(pi)] Before an "inverse function" - can be defined as an "inverse function" - what MUST happen = the inverse function's graph must pass the "vertical line test" AND the "horizontal line test" - meaning: you can draw a vertical and a horizontal line anywhere on the graph, if that line hits the graph in more than one spot, then the graph is NOT an inverse function - if NOT = it has passed & is an inverse function Define the BIGGEST difference between a "function" and "inverse function" = (function) = D(domain) & R(range) = (inverse function) = D(range) & R(domain) Explain what this would look like if you sketched - [y = -2sin(3x-120(degrees))] - [y = -2sin(3x-120(degrees))] - the [A] - "amplitude/height" of the curve will be = (2) - however, since (2) is "negative" the curve will be - "upside down" - the period of this sketch can be found by taking (360/3 (degrees)) = 120 (degrees) - since we see parenthesis within the problem - we know there will be a "left-right shift" - to solve we first set the entire expression in the parenthesis = to zero - (3x-120 degrees = 0 degrees) we can do this because - (sin0 = 0 degrees) - after solving we get - (x) = 40 (degrees) = this means that a "left-right shift" of y=0 is 40 (degrees) to the right since 40 is positive - since nothing is being added or subtracted after the parenthesis - we can conclude there is NO "up-down shift" = (D) Solving for the Trig Value - using graphing : [cos150 (degrees)] 1. draw a coordinate plane - (x and y-axis) 2. since - 150 (degrees) is located in the 2nd quadrant - so we will draw a right-triangle in the 2nd quadrant 3. then by starting at the positive side of the (x-axis) as 0 (degrees) - the angle will be formed by ending in (Q2) at 150 (degrees) - this angle is defined as = (angle (Greek letter) "theta") 4. Since (Q1) = 90 (degrees) we can subtract - 150-90 = which equals 60 (degrees) - since (Q2) also = 90 (degrees) we then subtract 90-60 and get = 30 (degrees) - so the acute angle nearest to the origin (0,0) will be defined as 30 (degrees) - so the side across from 30 (degrees) will equal = (1) and the other "leg" of the triangle will have a side value of = (-square root 3) - since (x) is negative in (Q2) and the hypotenuse = (2) 5. think back to "basic trig functions" where (cos) = (x/r) - which in this case would be = [(-square root 3)/2] 6. the solution - (cos)150 (degrees) = [(-square root 3)/2] Inverse sin (0) = = 0 Inverse sin (1) = = 1/2(pi) (angle theta) = (inverse cos) A : (A) = Domain (x) (inverse cos) A - (can have a domain) = within the #'s between (-1) and (1) ** [ (inverse cos) A ] & [ (inverse sin) A ] : = both have the same domain Inverse sin (-1) = = -1/2(pi) (angle theta) = (inverse sin) A : Define (A) and what (A) can equal (A) = Domain (x) (inverse sin) A - (can have a domain) = within the #'s between (-1) and (1) ** [ (inverse cos) A ] & [ (inverse sin) A ] : = both have the same domain (angle theta) = (inverse tan) A : (A) = Domain (inverse tan) A - can = (any real #) Another "notion" for inverse sin than = sin^(-1) is = = (sin arc A) How many answers can - (inverse sin) - have = ONLY (1) answer Define the 8 Trig Identities: ... All solutions to Trig Equations will be between what (#'s) = (degrees) = (0-360) = (radians) = (0 - 2(pi)) ** including (0) zero When solving a Trig Equation for (x) what should you remember to do first = First = FACTOR = Second = (set each factored item = 0) = Third = (solve for x) 5 sandwiches, 4 desserts and 3 drinks - (you can only have 1 of each) - how many different meals can you chose from = (5)(4)(3) = 60 Combinations vs. Permutations = combinations - order does not matter = permutations - order does matter - think (p) = "position" Example of - Permutations w/ NO repetition - In the lock above, there are 10 numbers to choose from (0,1,2,3,4,5,6,7,8,9) and we choose 3 of them: = 10 × 10 × ... (3 times) = 103 = 1,000 permutations When "graphing a linear inequality" what should you do first before even graphing = moving (y) to the left hand side & everything else to the right Define the 3 MAIN Steps for "graphing a linear inequality" 1. plot the "y=" line 2. if (y</equal to), or (y>/equal to) = the line will be SOLID on the graph if (y<), or (y>) = the line will be DASHED on the graph 3. shade ABOVE the line if "(y> or y>/equal to)" shade BELOW the line if "(y< or y</equal to)" Define & Explain - the 2 Types of Permutations 1. repetition - (a locks code = 333) - (n^r) : (n = number of things to choose from/possibilies) (r = how many you choose at a time) 2. NO repetition - (3 people are running a race, one person can not be first and second) - n!(n − r)! *Instead of writing the whole formula, people use different notations such as these: - permutation notation P(n,r) = nPr = n!/(n-r)! Solve: 8^(-2) - 8^(-2) = 1/8^(2) = 1/64 Define "reciprocal" of 8 = 1/8 Define "reciprocal" of (x/y) = (y/x) Define "reciprocal" of (x/x+1) = (x+1/x) Define "reciprocal" of (x) = (1/x) or (x^-1) "radical" over (x^2) is the same as = x^(p/r) - (p) = power (2) - (r) = root (2) =(x) What is x in log3(x) = 5 = 3^5 = 243 log(base)x = (y) (x) = (base)^(y) loga(m^r) = - base = (a) = r ( loga m ) - base = (a) Define in terms: loge(x) - base = (e) = natural log - more commonly seen as = ln(x) ln(e^x) = = (x) e^(lnx) = = (x) If given: log(x) what is the "common base" used = 10 = log10(x) Change into fraction form: loga(x) = (logb(x)/logb(a)) - (b) = common base - 10 - (x) = UP - (base "a") = DOWN State the equation of an ellipse ... State the equation of a circle & define all variables r^2 = (x-h)^2 + (y-k)^2 (r) = radius (h,k) = the "center" of the circle at (x,y) State the equation of an parabola ... State the equation of an hyperbola ... If an expression = [4^(x+4)] - how do we lower the exponent so we can solve ... If a question asks for the inverse function of a function - how do we solve = switch the variables (x) and (y) and solve for (y) If a translation moves the graph of f(x) h units to the LEFT - (h) would = = (h>0) - "positive" # - comparing graphs (x=y) and (x = y+1) If a translation moves the graph of f(x) h units to the RIGHT - (h) would = = (h<0) - "negative" # - comparing graphs (x=y) and (x = y-1) State the arithmetic sequence formula & define all variables [an = a1 (n-1)d] a1 = the first term in the sequence an = the nth term at n position of series d = difference