Algebra 2A Final Exam

imaginary numbers

"The imaginary number" (i) was created so that equations with no real solutions could be "solved." i = √-1 i² = -1 i³ = -i i⁴ = 1

Equation of a circle with center (h,k) and radius r

(x-h)²+(y-k)²=r²

degrees definition and how to convert between degrees and radians

A measure for angles. There are 360 degrees in a full rotation.

definition of relation+function

A relation is a set of ordered pairs. A function is a relation in which every x (input has exactly one y (output).

inverse relations

A relation is a set of ordered pairs. The inverse relation is the set of ordered pairs obtained by exchanging the coordinates of each ordered pair. The domain of a relation becomes the range of its inverse, and the range of the relation becomes the domain of its inverse. translated over line y=x

extraneous solution

A root of a transformed equation that is not a root of the original equation because it was excluded from the domain of the original equation. Always check for extraneous solutions. Any solution that makes a denominator equal to zero is extraneous.

vertical line test

A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point.

"infinitely many solutions" vs "all reals"

All real numbers do not have imaginary numbers. (not completely sure if this is the complete definition lemme know if u know)

irrational numbers

The decimal form of an irrational number neither terminates nor repeats. Square roots of numbers that are not perfect squares are irrational numbers.

Heron's formula

Use to find the area of a triangle when you know SSS. where s = semi-perimeter of the triangle

continuous (compounding formula)

remember to put rate as a decimal if it is a percent

SOHCAHTOA

sin=opposite/hypotenuse cosine=adjacent/hypotenuse tan=opposite/adjacent

roots/zeros

synonyms: roots, solutions, zeros Real zeros show up as x-intercepts but imaginary zeros do not. Odd-degree functions will always have an odd number of real zeros. The number of times a graph crosses the x-axis equals the number of real zeros.

Change of base formula

when a log is written with no base, the base is equal to ten (this is called a common logarithm)

Factoring: GCF, grouping, difference of squares, sum/difference of cubes, deFOIL

If there are 3 terms: deFOIL 2n²+14n-16 2(n²+7n+8) 2(n+8)(n-1) If the leading coefficient and the last term are both large consider splitting the middle term.

even-even-odd rule for radicals

If a real variable to an even exponent is under a radical with an even index and, when the radical is eliminated, the resulting exponent on the variable is odd, then absolute value signs must be placed around the variable. (All numbers to which "exponent" and "index" refer are natural numbers.)

permutations vs. combinations

If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. on calc: MATH → PROB (2+3)

domain/range

In its simplest form the domain (x) is all the values that go into a function, and the range (y) is all the values that come out.

inverse functions

Inverse functions are functions that undo one another The functions are inverses if and only if f(g(x))=x and g(f(x))=x The inverse of f(x) is written as f⁻¹(x) and pronounced f inverse of x. translated over line y=x

polynomial long division

Polynomial long division is the process for dividing a polynomial by a binomial. *You need 0 placeholders for any missing term!* At the end of your answer you can add the remainder over the divisor. So for the problem shown, your final answer would be 3x²+9x+29-76/3

rational numbers

Rational numbers can be expressed as a ratio a/b, where a and b are integers and b is not zero. The decimal form of a rational number is either a terminating or repeating decimal. Integers (...,-3,-2,-1,0,1,2), whole numbers (0,1,2,3,4,...), and natural numbers (1,2,3,4,...) are subsets of rational numbers because every integer is equal to n/1.

real numbers

Real numbers consist of irrational and rational numbers.

synthetic division

Synthetic division is a short-hand version of long division. Remember to use placeholder 0s anywhere you have a missing term and to add down and multiply in a V. 1. Write the coefficients of the dividend so that the degrees of the terms are in descending order. 2. Write the constant r of the divisor x-r in the box. 3. Bring the 1st coefficient down. 4, Multiply the 1st coefficient by r, and write the product under the 2nd etc.

linear regression

Technique for finding the straight line that best-fits the values of a linear function, plotted on a scatter graph as data points. If a "line of best fit" is found, it can be used as the basis for estimating the future values of the function by extending it while maintaining its slope. On calculator: 1. Go to STAT PLOT (y=) and turn on Plot1. 2. Go to List (2nd STAT) and type in x and y values. 3. Go to STAT → CALC 4 and press calculate TADA

Law of Cosines

The LoC is a generalization of the Pythagorean Theorem, and works to find angles and side lengths of both right and non-right triangles. Use when you have SAS or SSS

Law of Sines

The LoS is a generalization of SOHCAHTOA, and works to find angles and side lengths of both right and non-right triangles. Use when you have SSA or ASA. To solve for any particular side or angle, you simply need one proportion (don't set all three equal).

radian definition (JMel said that this would be on the final as an open definition.)

The measure of a central angle of a circle (of radius r) that intercepts an arc of length r.

reciprocal

The reciprocal of x is 1/x whereas the inverse of 3/5 is 5/3.

exponential growth/decay problems that involve a rate of change

This formula describes the amount of substance, y, remaining after t (units of time) growing/decaying at a rate of r (you must convert r to a decimal; negative r is used for decay).

SAS Area Formula

This formula is derived from the fact that the height of the altitude in triangle ABC with base a is bsinC. Use when you have SAS of a triangle.

general form for exponential growth and decay

a = initial value = y-int b = growth/decay rate If the equation is decay, 0<b<1

distance/rate/time formula

distance = rate • time

vertex/center

the middle point of a circle or sphere, equidistant from every point on the circumference or surface

work problem formula

trick for doing work problems: you have to think of the problem in terms of how much each person/machine/whatever does work in a given unit of time. If A completes 1/x jobs per hour, and B completes 1/y jobs per hours, then, when working together, they can complete 1/y+1/x jobs per hour. Take the inverse of how many jobs they can complete per hour, which will give you how long it will take for them to complete one job

amplitude and period trig functions

y= a sin bx + k a is amplitude 2π/b = period k= vertical shift make a 5 point x/y chart to plot

inverse variation

y=k/x y varies inversely as x if there is some nonzero constant k such that xy=k or y=k/x, where x and y are not zero

direct variation

y=kx y varies directly as x if there is some nonzero constant k such that y=kx, k is called the constant of variation

joint variation

y=kxz y varies jointly as x and z if there is some nonzero constant k such that

Probabilities: "or" vs "and"

you multiply with and and divide with or