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**All ALEKS Math Answers [K-12 to High ED]**

Yes, we will be discussing all ALEKS Math topics such as Basic Math, Math placement test Q&A, and all previously asked questions in ALEKS for the grades K-12 to High ED.

### ALEKS Basic Math Answers

**Q. **Improper fraction**Ans:** A fraction whose numerator is larger than the denominator.

**Q.** Expanded form**Ans:** The expanded form of a number shows the sum of its place values.

**Q.** Commutative Property**Ans:** The order in which numbers are added or multiplied does not change the sum or product: a + b = b + a

**Q.** Identity Property of Addition**Ans:** If you add zero to a number, the sum is the same as that number: a + 0 = a

**Q.** Associative Property of Addition**Ans:** Changing the grouping of three or more addends does not change the sum: (a + b) + c = a + (b + c)

**Q.** GCF**Ans:** Greatest Common Factor: The greatest number that is a factor of two or more given numbers.

**Q.** Factor**Ans:** A whole number is a factor of another when it divides that number exactly

**Q.** Simplest form of a fraction**Ans:** A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

**Q.** Reciprocal**Ans:** The multiplicative inverse of a number. A flipped fraction pair: 1/4 x 4/1 = 1

**Q.** Multiplicative inverse**Ans:** A reciprocal of a number that is multiplied by that number resulting in a product of 1: A flipped fraction pair: 1/4 x 4/1 = 1

**Q.** Parallel**Ans:** Two lines are parallel if they never cross.

**Q.** Perpendicular**Ans:** Two lines are perpendicular if they cross at a right angle.

**Q.** Area**Ans:** The area of a flat figure is the number of square units needed to cover the figure: Length x Width

**Q.** Perimeter**Ans:** The distance around a figure.

**Q.** Evaluate**Ans:** find the value of

**Q.** LCM**Ans:** Least Common Multiple: The smallest number is a multiple of two or more given numbers.

**Q.** Mixed number**Ans:** A number made up of a whole number and a fraction.

**Q.** Whole number**Ans:** All positive, non-decimal, non-fractional integers.

**Q.** Integer**Ans:** All whole numbers (both positive and negative) and zero.

**Q.** Proper fraction**Ans:** A fraction with a numerator smaller than the denominator.

**Q.** Signed numbers**Ans:** Numbers that are either positive or negative.

**Q.** Algebraic expression**Ans:** An expression or formula with numbers, variables, and operations.

**Q.** Product**Ans:** The answer to a multiplication problem

**Q.** Sum**Ans:** The result of addition.

**Q.** Quotient**Ans:** The result of division

**Q.** Dividend**Ans:** The number that is being divided

**Q.** Divisor**Ans:** The number by which another number is divided.

**Q.** Remainder**Ans:** The amount left over when one number is divided by another.

**Q.** Difference**Ans:** The result of subtraction

**Q.** Minuend**Ans:** A number from which another number is subtracted.

**Q.** Subtrahend**Ans:** A number that is subtracted from another number

**Q.** Addend**Ans:** Any number is being added.

**Q.** Order of Operations**Ans:** Parentheses, exponents, multiplications & divisions (from left to right), additions & subtractions (from left to right).

**Q.** PEMDAS**Ans:** Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

**Q.** Acute angle**Ans:** Less than 90 degrees

**Q.** Obtuse angle**Ans:** An angle between 90 and 180 degrees

**Q.** Right angle**Ans:** 90-degree angle

**Q.** Identity Property of Multiplication**Ans:** If you multiply a number by one, the product is the same as that number: a x 1 = a

**Q.** Associative Property of Multiplication**Ans:** Changing the grouping of three or more factors does not change the product. (ab)c=a(bc)

**Q.** Zero Property of Multiplication**Ans:** The product of any number and 0 is 0

**Q.** Commutative Property of Multiplication**Ans:** Changing the order of the factors does not change the product: ab=ba

**Q.** Denominator**Ans:** The bottom number in a fraction

**Q.** Nominator**Ans:** The top number in a fraction.

**Q.** Mode**Ans:** The value that occurs most frequently in a given data set.

**Q.** Range**Ans:** It is the difference between the largest and smallest values in a data set. l – s = r

**Q.** Ratio**Ans:** It is a comparison of two quantities, often expressed as a fraction or a quotient.

**Q.** Volume of a solid (a 3-dimensional object)**Ans:** It is the amount of space it takes up. measured in cubic units.

**Q.** Cubic units of measure**Ans:** An object’s length x width x height; measures an object’s volume or capacity.

**Q.** Compute**Ans:** To determine by mathematics or calculation.

**Q.** Power (of a number)**Ans:** Exponent; the number of times a base number is multiplied by itself.

**Q.** Exponent**Ans:** How many times the base is used as a factor in the power of a number.

**Q.** 2 to the fourth power**Ans:** 2^4

**Q.** The fourth power of 2**Ans:** 2^4

**Q.** Powers of ten**Ans: **10^1, 10^2, 10^3, etc… the exponent in the power of 10 gives the number of zeroes.

**Q.** Power of 6**Ans:** 6^x

**Q.** Fourth power of 10**Ans:** 10’000

**Q.** Order of grouping symbols in algebra**Ans:** {[()]}

**Q.** Supplement angle**Ans:** One angle of a pair whose measures add up to 180

**Q.** Supplementary**Ans:** Two angles that add up to 180 degrees

**Q.** Linear pair**Ans:** A pair of adjacent angles whose non-common sides are opposite rays; any pair of linear angles are supplementary.

**Q.** Opposite rays**Ans:** Two rays that have a common endpoint and form a line.

**Q.** Ray**Ans:** A straight line extending from a point

**Q.** Adjacent**Ans:** Near, next to, adjoining.

**Q.** Adjoining**Ans:** Next to or adjacent.

**Q.** Linear Pair Property**Ans:** If two angles form a linear pair, then they are supplementary

**Q.** Justification of the Linear Pair Property**Ans:** Verifying the truth of the Linear Pair Property by adding the two angles. By the angle addition property m∠1m∠2m∠ABC, and by definition, m∠ABC180°. Thus, m∠1m∠2180°, and the angles must be supplementary.

**Q.** Angle Addition Property**Ans:** Two or more adjacent angles can be added together to create a single larger angle

**Q.** Angle**Ans:** A figure formed by two rays with a common endpoint

**Q.** Vertex angle**Ans:** The angle is formed by the legs of an isosceles triangle.

**Q.** Isosceles triangle**Ans:** A triangle with at least two congruent sides

**Q.** Congruent**Ans:** Having the same size and shape

**Q.** Vertex of a polygon**Ans:** A point at which two sides of a polygon meet. The plural of vertex is vertices.

**Q.** Polyhedron**Ans:** A three-dimensional figure with faces that are polygons

**Q.** Faces**Ans:** The flat surfaces of a 3-D figure

**Q.** Vertex of a polyhedron**Ans:** A point where three or more edges meet

**Q.** Slant height**Ans:** The height of each lateral face

**Q.** Lateral face**Ans:** In a polyhedron, a face that is not a base.

**Q.** Polyhedron base**Ans:** The face that is not a lateral face

**Q.** The altitude of a triangle**Ans:** A perpendicular segment from a vertex to the line containing the opposite side

**Q.** The altitude of a polyhedron**Ans:** A perpendicular segment from the base to the opposite vertex or face; height

**Q.** Complementary angles**Ans:** Two angles whose sum is 90 degrees

**Q.** Complement angle**Ans:** One angle of a pair whose sum is 90 degrees

**Q.** Vertical angles**Ans:** A pair of opposite congruent angles formed by intersecting lines

**Q.** Distribute property**Ans:** Multiplying a number by a sum or difference in parentheses.

**Q.** Prime number**Ans:** A whole number that has exactly two factors, 1 and itself.

**Q.** LCD**Ans:** Least Common Denominator is defined as the least common multiple of the denominators of two or more fractions.

**Q.** Sum of the angles in a triangle**Ans:** 180°

**Q.** Squared Number**Ans:** A number multiplied by itself or a number with an exponent of 2

**Q.** Square root**Ans:** One of two equal factors of a number

**Q.** Scalene triangle**Ans:** A triangle with no congruent sides; no sides (or no angles) have the same measure

**Q.** Equilateral triangle**Ans:** A triangle with three congruent sides or all sides (or all angles) has the same measure.

**Q.** Divisibility rule for 3**Ans:** A whole number is divisible by 3 when the sum of its digits is divisible by 3.

**Q.** Divisibility rule for 9**Ans:** A whole number is divisible by 9 when the sum of its digits is divisible by 9.

**Q.** If a is positive, then a^n is?**Ans:** Positive

**Q.** If a is negative and n is odd, then a^n is?**Ans:** Negative

**Q.** If a is negative and n is even, then a^n is?**Ans:** Positive

**Q.** Terminating decimal**Ans:** A decimal whose digits end

**Q.** Repeating decimal**Ans:** A decimal that repeats a digit or group of digits forever.

**Q.** 1 foot = ? inches**Ans:** 12

**Q.** Prime factorization**Ans:** Breaking down a composite number until all of the factors are prime

**Q.** Area of a circle**Ans:** A=πr²

**Q.** Pythagorean Theorem**Ans:** a²+b²=c²

**Q.** Hypothenuse**Ans:** The side of a right triangle opposite the right angle

**Q.** Legs of a right triangle**Ans:** The two sides that form the right angle

**Q.** Parallelogram**Ans:** A quadrilateral with two pairs of parallel sides

**Q.** Quadrilateral**Ans:** A four-sided polygon

**Q.** Polygon**Ans:** A closed plane figure made up of line segments

**Q.** Trapezoid**Ans:** A quadrilateral with exactly one pair of parallel sides

**Q.** Area of a parallelogram**Ans:** A=bh

**Q.** Area of a trapezoid**Ans:** A=1/2h(b1+b2)

**Q.** Volume of a cylinder**Ans:** V=πr²h

**Q.** Coefficient**Ans:** A number multiplied by a variable in an algebraic expression.

**Q.** Interval notation**Ans:** A notation for describing an interval on a number line. The interval’s endpoint(s) are given, and a parenthesis or bracket is used to indicate whether each endpoint is included in the interval.

**Q.** Standard notation**Ans:** A number is written without exponents: an integer or a decimal fraction: 2.96 x 10^4= 29600

**Q.** Compound inequality**Ans:** Two or more inequalities joined together by “and” or “or”

**Q.** Rational numbers are**Ans:** Fractions

**Q.** Irrational numbers**Ans:** Numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.

**Q.** 1 oz = *_* grams**Ans:** 28.4 grams

**Q.** 1 liter = *__* pints**Ans:** 2.1 pints

**Q.** 1 mile = *__* kilometers**Ans:** 1.6 km

**Q.** 1 inch = *_* cm**Ans:** 2.54 cm

**Q.** Circumference defined as**Ans:** Distance around the circle

**Q.** Circumference formula**Ans:** C=2πr

**Q.** Area of a rectangle**Ans:** A=lw

**Q.** Area of a triangle**Ans:** A=1/2bh

**Q.** Area of a circle**Ans:** A=πr²

**Q.** Area of a square**Ans:** A=s²

**Q.** Volume of a Pyramid/Cone**Ans:** v=1/3Bh

**Q.** Volume of a cylinder**Ans:** V=Bh

**Q.** Volume of a sphere**Ans:** 4/3πr³

**Q.** Integers are**Ans:** Positive and negative whole numbers and zero

**Q.** Consecutive even integers**Ans:** Even numbers that follow each other on a number line ie: −6,−4,−2,0… variable form: n, n+2, n+4, n+6

**Q.** Consecutive odd integers**Ans:** Odd numbers that follow each other on a number line ie: −5,−3,−1,1… variable form: n, n+2, n+4

**Q.** Interest earned**Ans:** Amount invested x rate = interest earned

**Q.** Distance**Ans:** rate x time

**Q.** “And” in combined inequalities**Ans:** Only looking for what they have in common

**Q.** “Or” in combined inequalities**Ans:** Includes everything mentioned

**Q.** When would you change the direction of an inequality?**Ans:** When you multiply or divide by a negative number.

**Q.** Associative Property of Addition**Ans:** (a+b)+c=a+(b+c)

**Q.** Zero property of addition**Ans:** Adding 0 to a number equals the original number

**Q.** Commutative Property of Addition**Ans:** a+b=b+a

**Q.** Identity Property of Addition**Ans:** a+0=a

**Q.** Absolute value**Ans:** The distance a number is from zero on a number line. ALWAYS POSITIVE, may represent units.

**Q.** Scalene triangle**Ans:** A triangle with no congruent (equal) sides

**Q.** Median**Ans:** the middle score in a distribution; half the scores are above it and half are below it

**Q.** Mean**Ans:** the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores

**Q.** An ordered pair**Ans:** Any pair of numbers for which order is important. Example: (x,y)

**Q.** Domain**Ans:** The set of x-coordinates in a relation.

**Q. **Relation**Ans:** A set of ordered pairs

**Q.** Range**Ans:** The set of all y-values

**Q.** It is okay to have a zero in the **_** of a fraction.**Ans:** Numerator

**Q.** You cannot have a zero in the **_** of a fraction.**Ans:** Denominator

**Q.** What is a function?**Ans:** Each x must be paired with only one y value

**Q.** Vertical line test**Ans:** If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.

**Q.** A function can have the same y coordinate, but not the same x coordinate.**Ans:** True

**Q.** Graphing Inequalities**Ans:** A solid line means the inequality is ≥ or ≤. A dotted line means the inequality is > or <.

**Q.** How do you know which side to shade when graphing inequalities?**Ans:** Create a test point (usually (0,0). If the resulting point is TRUE, Shade the side that includes the test point.

**Q.** 4 basic types of transformations**Ans:** Reflection, rotation, translation, dilation

**Q.** For any flat line, the slope is always:**Ans:** m=0

**Q.** For any vertical line, the slope is always:**Ans:** Undefined, no slope

**Q.** Slope formula**Ans:** m=y2-y1/x2-x1

**Q.** Slope-intercept form**Ans:** y=mx+b, where m is the slope and b is the y-intercept of the line.

**Q.** Commutative Property**Ans:** The property says that two or more numbers can be added or multiplied in any order without changing the result.

**Q.** Associative Property**Ans:** The way in which numbers are grouped does not change their sum or product

**Q.** Identity Property of Addition**Ans:** If you add zero to a number, the sum is the same as that number.

**Q.** Velocity formula**Ans:** v=d/t

**Q.** Factoring formula for the difference of squares**Ans:** A^2 – B^2 = (a+b)(a-b)

**Q.** Rewrite the following without an exponent:**Ans:**

(-4)^-3

A^-n=1/a^n

-1/64

**Q.** Quadratic Formula**Ans:** -b±[√b²-4ac]/2a

**Q.** Slope**Ans:** (y₂-y₁)/(x₂-x₁)

**Q.** Slope-Intercept**Ans:** y=mx+b

**Q.** a³-b³**Ans:** (a-b)(a²+ab+b²)

**Q.** a³+b³**Ans:** (a+b)(a²-ab+b²)

**Q.** a²-b²**Ans:** (a-b)(a+b)

**Q.** a²-2ab+b²**Ans:** (a-b)²

**Q.** a²+2ab+b²**Ans:** (a+b)²

**Q.** (a+b)(c+d)**Ans:** ac+ad+bc+bd

**Q.** a(b+c)**Ans:** ab+ac

**Q.** sine ratio**Ans:** opposite ÷ hypotenuse

**Q.** cosine ratio**Ans:** adjacent ÷ hypotenuse

**Q.** tangent ratio**Ans:** opposite ÷ adjacent

**Q.** Direct Variation**Ans:** y=kx

**Q.** Inverse Variation**Ans:** y=k/x

**Q.** Point-Slope form**Ans:** y-y₁=m(x-x₁)

**Q.** Standard form**Ans:** Ax + By=C, where A, B, and C are not decimals or fractions, where A and B are not both zero, and where A is not a negative

**Q.** Undefined**Ans:** When there is a vertical line that has different y points, but the same x point

**Q.** Zero**Ans:** When there is a horizontal line that has different x points, but the same y point

**Q.** Dividing by a negative number in an inequality**Ans:** You must flip the sign

**Q.** Graphing < or > on a coordinate plane**Ans:** Dotted line

**Q.** Graphing ≥ or ≤ on a coordinate plane**Ans:** Solid line

**Q.** Graphing ≥ or > on a coordinate plane**Ans:** Shade upwards or to the right

**Q.** Graphing ≤ or < on a coordinate plane**Ans:** Shade downwards or to the left

**Q.** Infinitely many solutions**Ans:** When the system of equations have the same slope and y-intercept

**Q.** One solution**Ans:** When the system of equations have different slopes

**Q.** No solution**Ans:** When the system of equations have the same slope but different y-intercepts

**Q.** Linear parent function**Ans:** y=x or f(x)=x

**Q.** Elimination method**Ans:** Solving systems by adding or subtracting equations to eliminate a variable

**Q.** Solution of the system of linear equations**Ans:** Any ordered pair in a system that makes all the equations true

**Q.** Graphing method**Ans:** Graphing the system of equations and finding the point at which they intersect

**Q.** Substitution method**Ans:** Replacing one variable with an equivalent expression containing the other variable

**Q.** Absolute value equation**Ans:** V-shaped graph that points upward or downward

**Q.** Translation**Ans:** A shift of a graph horizontally, vertically, or both, results in a graph of the same shape and size, but in a different position.

**Q.** Area of a circle**Ans:** Πr²

**Q.** Area of a square**Ans:** s², where, s = length of a side

**Q.** Area of a triangle**Ans:** ½(base x height) [or (base x height)÷2]

**Q.** Area of a trapezoid**Ans:** ½(b₁ +b₂) x h [or (b₁ +b₂) x h÷2]

**Q.** Perimeter of a rectangle**Ans:** 2Length + 2width [or (length + width) x 2]

**Q.** Perimeter of a square**Ans:** 4s (where, s = length of a side)

**Q.** Area of rectangle, square, parallelogram**Ans:** A=bh

**Q.** Area of a sector**Ans:** x°/360 times (∏r²), where x is the degrees in the angle

**Q.** length of a sector**Ans:** x°/360 times (2 pi r), where x is the degrees in the angle

**Q.** Circle**Ans:** Is the set of points that are all the same distance (its radius) from a certain point( the center).

**Q.** Radius (Radii)**Ans:** A segment connecting the center of a circle to any point on the circle

**Q.** Diameter**Ans:** The distance across the circle through the center of the circle. The diameter is twice the radius.

**Q.** Chord**Ans:** The distance from one point on the circle to another point on the circle.

**Q.** Sector**Ans:** The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle.

**Q.** Arc**Ans:** Part of a circle connecting two points on the circle.

**Q.** Central Angle**Ans:** An angle whose vertex is the center of the circle

### ALEKS Math Placement Test Answers

Below you can find answers for ALEKS Math Placement test questions that we collected from past 4 years to till now:

**Q.** Positive Number**Ans:** Any number greater than zero

**Q.** Negative Number**Ans:** Any number less than zero

**Q.** Even Number**Ans:** Any integer divisible by 2

**Q.** Odd Number**Ans:** Any integer not divisible by 2

**Q.** Sum**Ans:** The result of two numbers being added

**Q.** Difference**Ans:** The result of one number being subtracted by another

**Q.** Product**Ans:** The result of two numbers being multiplied

**Q.** Quotient**Ans:** The result of one number being divided by another

**Q.** Factor**Ans:** A number that is multiplied to form the product

**Q.** Greatest Common Factor**Ans:** The greatest whole number that is a factor of two or more numbers

**Q.** Multiples**Ans:** The product of a number and a counting number

**Q.** Least Common Multiple**Ans:** The least whole number that is a multiple of two or more numbers

**Q.** Prime**Ans:** A number greater than one whose only whole number factors are one and itself

**Q.** Composite**Ans:** A number greater than one with more than two whole-number factors

**Q.** Real**Ans:** All positive and negative numbers and zero

**Q.** Rational**Ans:** Any number that can be expressed as the quotient of two integers

**Q.** Irrational**Ans:** Real numbers that are not rational

**Q.** How to find Percent Increase or Decrease**Ans:** Difference/Original = Percent Increase or Decrease

**Q.** Average Problems**Ans:** Average x Number = Total

**Q.** Simple Probability Problems**Ans:** Outcomes giving desired result/Total possible outcomes

**Q.** Conditional Probability**Ans:** The probability of an event, given another event, has already occurred. Ex: The probability that a playing card from a standard deck is a 5, given that it is red, is 2/26.

**Q.** Joint Probability**Ans:** The probability of two events occurring at the same time. Ex: The probability that a playing card from a standard deck is red and a 5 is 2/52.

**Q.** graph ≥, ≤**Ans:** solid line

**Q.** graph >, <**Ans:** DOTTED line

**Q.** every, each, per (writing equations)**Ans:** rate/ set y value equal to (multiply this value by the x variable)

**Q.** When writing an equation or working ANY inequality problem what does it help to do first?**Ans:** Write out what’s happening

**Q.** What should you double-check for EVERY time**Ans:** Writing problem, negative signs, uncarried negative signs, unflipped inequality signs, undotted lines, incorrect multiplication, fully simplified

**Q.** distance =**Ans:** rate x time

**Q.** When solving a system of equations can you add the equations together if one of the variables will cancel out?**Ans:** YES

**Q.** What should you always write out when doing your work?**Ans:** Fraction conversions, simplification steps, line graphs (interval notation)

**Q.** What standard does an equation/ graph have to meet to be considered a function?**Ans:** Each input can only have one output

**Q.** How do you translate a graph up or down?**Ans:** The value outside of the parentheses (DONT change sign)

**Q.** How do you translate a graph side to side?**Ans:** The value inside the parentheses (CHANGE SIGN)

**Q.** Translate -(x + 3)^3 – 2 from -x^3**Ans:** left 3, down 2

**Q.** When translating do you need to solve the equation?**Ans:** No, just look at the numbers inside (side to side change sign) and outside (up and down DONT change sign) the parentheses

**Q.** U represents**Ans:** Numbers in EITHER set (Union)

**Q.** ⋂ represents**Ans:** Numbers in BOTH sets (Intersection)

**Q.** What is the quadratic formula?**Ans:** -b±√b^2-4ac/2a

**Q.** (x-6)(x+4) < 0**Ans:** Where is the product of (x-6) and (x+4) less than 0?

**Q.** Should you graph two separate equations on the same number line when doing your work?**Ans:** No

**Q.** When you take the square root of a variable, how many answers are there?**Ans:** 2 answers (negative and positive)

**Q.** x^a/x^b**Ans:** x^a-b

**Q.** x^a times x^b =**Ans:** x^a+b

**Q.** (x^a)^b**Ans:** x^ab

**Q.** What should you always physically mark on inequality graphs that are getting compared before attempting to solve the problem?**Ans:** Where the graphs intersect

**Q.** BETC**Ans:** Divide the top leading coefficient by the bottom leading coefficient (Bottom Equals Top Coefficient HORIZONTAL ASYMPTOTE)

**Q.** BOB0**Ans:** The horizontal asymptote = 0 (Bigger On Bottom 0)

**Q.** BOTU**Ans:** There is no horizontal asymptote, find slant asymptote (Bigger On Top Undefined)

**Q.** How do you find the slant asymptote?**Ans:** Divide the ENTIRE numerator by the ENTIRE denominator, the slant asymptote is the quotient, ignore the remainder (might include division with variables raised to exponents)

**Q.** How do you find the vertical asymptote?**Ans:** Set the denominator equal to 0

**Q.** sin**Ans:** csc

**Q.** cos**Ans:** sec

**Q.** tan**Ans:** cot

**Q.** Can a graph have more than one local maximum or minimum?**Ans:** yes

**Q.** volume**Ans:** cubed

**Q.** area**Ans:** squared

**Q.** perimeter/ length**Ans:** no power

**Q.** How do you find the vertex of a parabola?**Ans:** Take the derivative (slope) and set it equal to 0

**Q.** d/dx (x^a)**Ans:** ax^a-1

**Q.** antiderivative (x^a)**Ans:** (x^a+1)/a+1

**Q.** area of a circle**Ans:** πr²

**Q.** Translate this equation to the right 3 and up 4:**Ans:** f(x)=1/2x^2(1/2(x-3)^2)+4

**Q.** If a question asks for all real zeroes should you include imaginary numbers?**Ans:** no

**Q.** In the functions (f + g)(x), (f-g)(1), (f・g)(x), (f/g)(2) etc. is the x or the coefficient in the second parentheses being multiplied by the solution of f and g?**Ans:** No, x is the input for the combined functions (just like it would be in single functions like f(x))

**Q.** y=af(x)**Ans:** multiply y coordinate by a

**Q.** y=f(ax)**Ans:** divide x coordinate by a

**Q.** to find x intercept**Ans:** set f(x) equal to 0

**Q.** to find y intercept**Ans:** find f(0)

**Q.** What are inverse functions.**Ans:** Functions that switch the input and output

**Q.** How do you find the inverse function of an equation?**Ans:** Solve for x then swap the variables of the new solved equation

**Q. **What should you always do first**Ans:** Cancel/ simplify/ factor

**Q.** When a problem says that one variable varies inversely with another variable what do you need to know/ find?**Ans:** the constant k

**Q.** Direct variation**Ans:** y=kx (ratio remains constant (y/x stays the same))

**Q.** Inverse variation**Ans:** y=k/x (product remains constant (yx stays the same))

**Q.** Joint variation**Ans:** y=kzx (y varies jointly with x and z (y/xz stays the same))

**Q.** When doing polynomial long division, which terms do you use**Ans:** The leading coefficients

**Q.** What do you normally confuse the least common multiple with?**Ans:** The greatest common factor (the least common multiple IS NOT the greatest common factor)

**Q.** How do you find the least common multiple?**Ans:** Find the smallest whole number that both expressions divide evenly into. Then use the variables with the highest power

**Q.** What is the least common multiple of 25u^3v^4 and 10w^2u^5v^6 ?**Ans:** 50u^5w^2v^6

**Q.** When performing operations on rational expressions, what does it help to circle?**Ans:** Subtraction signs that apply to an entire expression (apply them at the very end)

**Q.** When should you find the Lowest Common Factor**Ans:** ANY TIME you are adding or subtracting fractions, regardless of whether or not there are variables

**Q.** When performing operations on rational expressions, if there is a coefficient, do you need to multiply the coefficient by the lowest common factor as well?**Ans:** YES

**Q.** Can you cancel out the denominator by multiplying both sides by its reciprocal regardless of whether or not there is a zero on the other side?**Ans:** YES

**Q.** Can a solution make the denominator equal 0?**Ans:** NO (ALWAYS CHECK FOR THIS BEFORE ANSWERING if a “solution” does this it is not a solution. it iS A FRAUD>~< )

**Q.** b√x^a**Ans:** x^a/b

**Q.** x^a/b**Ans:** b√x^a

**Q.** What is the vertex of this equation?**Ans:** -3(x-5)+2 (the parentheses are absolute val. symbols) (5,2)

**Q.** When you’re trying to find the width do you disregard negative answers?**Ans:** yes

**Q.** When the number under the radical is large should you trust that it’s simplified fully?**Ans:** No (it’s probably not fam)

**Q.** What should you do any time you solve by squaring?**Ans:** Plug the answers back in (to double-check)

**Q.** How do you find the domain of an equation with a radical?**Ans:** Set the value under the radical equal to 0. IGNORE values outside the radical. (The domain ONLY includes values that make the numbers under the radical greater than or equal to 0)

**Q.** What helps when you’re trying to simplify a radical with a power higher than 2?**Ans:** Writing out a table with perfect roots of the power

**Q.** What is the range of an inverse function?**Ans:** The domain of the original function

**Q.** When finding the domain of composition functions should you find the domain of the simplified function or the unsimplified function?**Ans:** The unsimplified function (ESPECIALLY if there are radicals)

**Q.** Should you leave negatives in the denominator?**Ans:** no (multiply by everything negative 1)

**Q.** log_a c = b if**Ans:** a^(b) = c

**Q.** log_6 36 = 2**Ans:** 6^2 = 36

**Q.** log**Ans:** log_10

**Q.** What do you do for exponential growth or decay problems?**Ans:** initial x percent^time

**Q.** log_a (xy)**Ans:** log_a x + log_a y

**Q.** log_a(x/y)**Ans:** log_a x – log_a y

**Q.** log_a x^r**Ans:** rlog_a x

**Q.** Should you round down when a problem asks for the smallest whole number?**Ans:** No (round up to the next whole number regardless of the decimal size- it can be less than .5 and you’d still round up)

**Q.** If two logs with the same base are on different sides of an equation can you cancel the logs?**Ans:** Yes (queen:3)

**Q.** Can you take the log of a negative number?**Ans:** NO

**Q.** What do you do when solving complex log/exponential equations?**Ans:** distribute and factor

**Q.** What is -log when graphing?**Ans:** A reflection about the x-axis (multiply y coordinates by negative 1)

**Q.** What do you need to change when graphing log transformations that involve a change in x coordinate?**Ans:** The domain and vertical asymptote

**Q.** What is the vertical asymptote of a normal log equation?**Ans:** the y axis

**Q.** How do you complete the square?**Ans:** Divide the middle coordinate by 2, square it (This is the third coefficient) , ADD THIS TO THE OTHER SIDE OF THE EQUATION AS WELL

**Q.** What is the equation of a circle**Ans:** (x-h)^2 + (y-k)^2 = r^2

**Q.** What is the center of the circle (x-h)^2 + (y-k)^2 = r^2**Ans:** (h,k) (h -> -h)

**Q.** What is the radius of the circle (x-h)^2 + (y-k)^2 = r^2**Ans:** r

**Q.** You graph negative angles in standard form**Ans:** clockwise

**Q.** You graph positive angles in standard form**Ans:** counterclockwise

**Q.** How do you find the value of an angle?**Ans:** Find its coordinate on the unit circle

**Q.** What are the Pythagorean identities?**Ans:**

sin^2 + cos^2 = 1

tan^2 + 1 = sec^2

cot^2 + 1 = csc^2

**Q.** Arc length formula**Ans:** s=r(theta)

**Q.** isosceles triangle**Ans:** two congruent sides

**Q.** polygon**Ans:** closed figure with no crossing / curved lines

**Q.** regular polygon**Ans:** all sides and angles are equal

**Q.** pentagon**Ans:** 5

**Q.** hexagon**Ans:** 6

**Q.** heptagon**Ans:** 7

**Q.** quadrilateral**Ans:** four sided polygon

**Q.** trapezoid**Ans:** one pair of parallel sides

**Q.** rhombus**Ans:** A parallelogram with four congruent sides

**Q.** rectangle**Ans:** A parallelogram with four right angles

**Q.** square**Ans:** A parallelogram with four congruent sides and four right angles

**Q.** central angle**Ans:** An angle whose vertex is the center of the circle

**Q.** chord**Ans:** A segment whose endpoints lie on a circle

**Q.** (3-d figures) face**Ans:** flat surface

**Q.** (3-d figures) edge**Ans:** intersection of two faces

**Q.** (3-d figures) vertex**Ans:** intersection of three edges

**Q. **prism**Ans:** 3-d figure with parallel / congruent faces

**Q.** volume**Ans:** The amount of space an object takes up

**Q.** volume equation**Ans:** lwh

**Q.** (volume formula) cube**Ans:** side^3

**Q.** (volume formula) triangle**Ans:** (1/2bh)h

**Q.** (volume formula) pyramid**Ans:** 1/3(h)(area of base)

**Q.** (volume formula) cylinder**Ans:** (pi)r^2h

**Q.** (volume formula) cone**Ans:** 1/3(pi)r^2h

** Q.** (volume formula) sphere

**Ans:**4/3(pi)r^3

**Q.** slope formula**Ans:** y2-y1/x2-x1

**Q.** no slope**Ans:** zero on bottom

**Q.** zero slope**Ans:** 0/6

**Q.** slope-intercept form**Ans:** y=mx+b

**Q.** point-slope form**Ans:** y-y1=m(x-x1)

**Q.** horizontal slope**Ans:** m=0

**Q.** vertical slope**Ans:** undefined

**Q.** direct variation formula**Ans:** y=kx

**Q.** how to: direct variation**Ans:** solve for k, plug k back into direct variation formula, plug in x / y values and solve

**Q.** vertex form**Ans:** y-k=a(x-h)^2, vertex = (h,k)

**Q.** (vertex form) y=ax^2**Ans:** vertex is (0,0)

**Q.** (vertex form) y=a(x-h)^2**Ans:** vertex is (h,0)

**Q.** (vertex form) y-k=ax^2**Ans:** vertex is (0,k)

**Q.** Axis of symmetry equation**Ans:** x=h

**Q.** Axis of symmetry**Ans:** The line that divides graph into two congruent reflected halves

**Q.** sum and product of roots**Ans:** 0=x^2-(sum of roots) + (products of roots)

**Q.** minimum value**Ans:** smallest y value

**Q.** quadratic formula**Ans:** x = -b ± √(b² – 4ac)/2a

**Q.** 45-45-90 triangle equal sides**Ans:** x

**Q.** 45-45-90 triangle hypotenuse**Ans:** (√2)x

**Q.** 30-60-90 triangle short side**Ans:** x

**Q.** 30-60-90 triangle-long side**Ans:** x√(3)

**Q.** 30-60-90 triangle hypotenuse**Ans:** 2x

**Q.** i^2**Ans:** -1

**Q.** i^3**Ans:** -i

**Q.** i^4**Ans:** 1

**Q.** distance formula**Ans:** d = √[( x₂ – x₁)² + (y₂ – y₁)²]

**Q.** midpoint formula**Ans:** (x₁+x₂)/2, (y₁+y₂)/2

**Q.** equation of a circle**Ans:** (x-h)²+(y-k)²=r²

**Q.** Permutation**Ans:** of ways something can be ordered

**Q.** permutation equation**Ans:** n!/(n-r)!

**Q.** combination**Ans:** A grouping of items in which order does not matter

**Q.** combination equation**Ans:** n!/r!(n-r)!

**Q.** change of base formula**Ans:** logaX=logbX/logbA

*Note: Math questions in ALEKS are prepared by mathematics professors whose areas of expertise give the program the ability to ask questions that are both challenging and beneficial to the learning process.*

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