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PRACTICE QUESTIONS

Geometry Topics: Triangle Centers
 * Algebra 2 Review**

Permutations/Combinations · nPr= n! / (n-r)!

· nCr= n! / r! (n-r)!

Radical Conjugate- in a+b, if one is a radical, a-b (ex. 1 / 2+root2, the radical conjugate would be the 2-root2 used to rationalize the denominator)

Function Forms · Linear Functions:

Slope-Intercept: y=mx+b m = - A / B b = C / B Standard: Ax + By = C o Point-Slope: y-y1 = m(x-x1)

· Circle (not a function): (x-h)2 + (y-k)2 = r2

· Absolute Value: y = a|x-h|+k

· Parabolas/Quadratics:

o Vertex Form: y = a(x-h)2 + k h = -b / 2a o Standard Form: y = ax2 + bx + c (h,k) is the vertex or center, if applicable b or __consummate__ (lowercase) c= y-intercept m = slope r = radius Discriminant: b2-4ac If > 0, 2 x-intercepts If = 0, 1 x-intercept (the vertex) If < 0, 0 x-intercepts (doesn't cross x-axis)

Other Formulas · Simple Interest:I = PRT Interest = Principle * Rate * Time · Compound Interest:A = P(1+r/n)nt Amount = Principal * (1 + Rate / Rate Compounded per Unit of Time, ex 360 if 1 per day and time is in years) ^ (Number Times Compounded * Time)

· Continuous Interest:A = Pert Amount = Principal * e ^ (Rate * Time) · Half-Life:T1/2 = ln.5 / k Half Life = Log e 0.5 / Constant Used for radioactive decay · General Exponential Growth/Decay:N = N0ekt

o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Final Amount = Initial Amount * e ^ (Constant * Time) <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">General model for decay or growth that's exponental (self explanatory, actually) · Distance/Time:D=RT <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Distance = Rate * Time

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Matrices · Addition/Subtraction: A+B=C <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Dimensions: mxn & mxn > mxn <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">First value = number of rows <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Second value = number of columns <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Add/Subtract Corresponding Values · Scalar Multiplication: kA = B <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Dimensions: constant & mxn > mxn <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Multiply all values by the constant · Multiplication AB=C <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Dimensions: mxn & nxp > mxp Inner dimensions __must__ be equal, outer ones determine result dimensions <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Multiply each row of A by each column of B, sum of each product determines cell in result (ex. [1 2] x [1 (imagine this is vertical) 4] = 1x1 + 2x4 = [9]) Notcommutative

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Determinant <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">|a b| <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">|c d| = ad-bc <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Essentially, multiply across diagonally down, then subtract the product from diagonally up <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Usually only for square matrices <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">For 3x3's, add first 2 columns again on the right, then multiply diagonals and add them up to make top and bottom totals <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">|A| = determinant of A · Cramer's Rule: AX=B <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">A is a 2x2 matrix, X is a 2x1 (tall- remember, 2 __rows__), [x y], B is a 2x1. <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Ax (imagine x is in subscript) = A, but the first column is swapped with B <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Ay = A, 2nd column swapped with B <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">x = |Ax| / |A| <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">y = |Ay| / |A| <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">If there's a z, make A 3x3 and X & B 3x1, then swap 3rd column for Az <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Can be used to solve systems of equations <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Take coefficients of x, y, and z; Plug them into A. <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Plug in x, y, and z down the X matrix <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Plug in what ax + by + cz equals (the constant, or non-variable) down the B matrix <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Find the determinants of A, Ax, Ay, and Az <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Divide, and you've solved for the variables!

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Singular Matrix
 * Determinant = 0

·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Identity Matrix: <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">[1 0] > <span style="color: #ffffff; display: block; font-family: Arial,sans-serif; font-size: 13pt;">........................[ <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">0 1], and so forth, with just a diagonal of 1's in a matrix of 0's. Interestingly, I think IA=A. · Inverse: A^-1 = 1 / |A| x [d -b] .................................................. .[-c a] o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">A * X = B is also X = A^-1 * B

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;"> Functions ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Composition- f(g(x)) · Inverse of a Function o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Reflect over y=x o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Aka, swap x & y in the function, keeping coefficients where they are o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Ex: f(x)=4x+2; f(x)-1 is x=4y+2, or y=-x/4 -1/2.

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;"> Linear Programming ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Constraints - Function Inequalities o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Ex: y>=x+2 ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Objective Function - Function that tests vertices for highest or lowest value o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Usually monetary, ex. to find best combination of 2 items to sell · What do you make axes represent? o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">The two different items you need as an ordered pair for the answer. · Example question <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">: o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">You are a yamaga and potakho farmer. You can plant a maximum of 100 yamagas or 200 potakhoes. You usually surround each yam with 4 potakhoes, so there needs to be at least 3 times as many potakhoes as yams. You can sell yamagas for a profit of $25 each, and potakhoes for a profit of $10 each. How many of each should you plant? o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Axes- Let's make the-x axis potakhoes and the y-axis yamagas. o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Constraints- x<=200; y<=100; y<=1/4x. o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Objective function: F(x,y)=25x+10y o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Vertices to plug in: § <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">(0,0) - Or not, to save time* § <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">(200,0) - All potakhoes § <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">(200,50) - Max potakhoes, then max yamagas sticking to y<=1/4x o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">*The Answer: Try to figure it out yourself! It's a bit tough to post its graph on the wiki. But it's a really easy problem (or is it?). Highlight the next line if you need help. § DOUBLE MEANING of the asterisk above! (0,0) IS the answer! Instead of planting, STEAL the yamagas and potakhoes of your hardworking next door neighbor and make a HUGE PROFIT! (Didn't mention yams sell for $100,025 and potatoes sell for $1,000,000,001, but they cost $100,000 and $999,999,991 to raise [do the math- profits of $25 and $10] - after all, they don't exist) Just joking, of course :-)

- Contains both real and imaginary parts, ex. 2+4i · i = i · i2 =-1 · i3 = -i · i4 = 1

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Polynomials · Must have whole number degrees per term (no fractions or negatives) · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Ex. 4x + y + 6 · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Not 2/x or xy-1 + x8 ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Degree - Greatest exponent of one of the terms ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Leading coefficient - coefficient of term with highest degree

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;"> Factoring Methods (in order of sequence) <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;"> 1. Greatest Common Factor 2. Difference of Squares 1. a2 - b2 = (a+b)(a-b) 3. Perfect Trinomial Squares 4. Grouping 5. Sum & Difference of Cubes o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">a3​ + b3 = (a+b)(a2-2ab+b2) o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">a3​ - b3 = (a-b)(a2+2ab+b2) 6. Trinomial Factoring (Guess + Check) 7. Completing the Square o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">x^2+ ax + a2/4 = # > (x+a/2) = root# 8. Quadratic Formula o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">(-b +/- root(b2 - 4ab) ) / 2a

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Logarithms · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">The

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Inverse <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">of the exponential functions (graph is reflected across y=x line, goes to a vertical asymptote) · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Essentially, logb m=x is equivalent to bx=m · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Poses question, "What exponent is needed to make this subscript number equal the second one?" o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">lnx = logex o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">logx = log10x o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">logbmn = logbm + logbn o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">logbm/n = logbm - logbn o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">logbmk = k*logbm o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">logblogbm = m · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt;">Change of base: logbm = logcm / logcb

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;"> Polynomial Division · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">If y=ax2+bx+c, sum of the roots = -b/a and product is c/a ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Synthetic Division <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">- when dividing by a linear binomial, put negative of constant in a box and multiply it by sum of digit and previous product o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Ex. x3-16x2+19x +396 / (x-9) o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Make leading coefficient 1 if it’s not, then fix change at the end o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">9| 1 -16 19 396 o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">....1...9.-63-396 o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">- o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">....1..7..-44…0 o <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Answer – x2 + 7x – 44 · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Factor theorem: if P(x) is polynomial, (x-a) is a factor of P(x) if P(a)=0 · <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Rational Root Theorem: If P(x) = a0xn+a1xn-l…+an, and p is a factor of an and q is a factor of a0, then p/q is a rational root Variation Between 2 Constants ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Direct Variation <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">(joint variation): x / y = k (k is a constant) ·

<span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">Inverse Variation <span style="display: block; font-family: Arial,sans-serif; font-size: 13pt; line-height: 115%;">: x*y = k

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