Math 2 Mod 1, 2, and 3

Multiplying two binomials

(X+P)(X+Q) Distribute out X(x+p)+Q(x+p) Distribute again Xx+XP+qx+pq Simplify X^2+(p+Q)x+pq

Multiplying complex numbers ( real scale factor)

3(2+5i) Use distributive property 6+15i Scales the vector by a factor of three on the complex plane. -3(2+5i)=-6-15i Rotates vector 180 degrees and scales by factor of three

Adding or subtracting complex numbers

Add the real parts together and the imaginary parts together. (4+5i)+(6-2i)=10+3i Use head tail method to add on complex plane.

Distance between complex numbers

It is the modulus of the difference between them. It is represented on the complex plane as the modulus of the vector that connects them

Finding stretch with factored form

MAKE SURE THERE ARE NO COEFFICIENTS OF X IN THE BINOMIALS. If factored form is y=a(x+p)(x+q) then multiply a by {1, 3, 5, 7...} For example if a = 4 then stretch is {4, 12, 20, 28...}

Simplifying exponents

Make sure there are no negative exponents and as few terms as possible. Distribute the exponents when possible. Combine like terms. (r^5)(s^3)/r(s^2) r^4(s^3)/s^2 (r^4)s r(f^-b) r/f^b

Representing numbers on the complex plane

Y axis is imaginary part, x axis is real part. Can be represented as vector or point.

Conjugate of a comlex number

complex number: a+bi conjugate: a-bi ____ Represented by a+bi Vector representation is a reflection of the complex number over the real axis. The sum of a complex number and its conjugate is always the real number a When a complex number is multiplied by its conjugate the answer is always the real number a^2+b^2

Exponents and radicals

m√a^n=a^(n/m) The mth root of a to the nth power is equal to a to the n/m power.

Rules of radicals

n√(ab) = n√a*n√b The nth root of ab is the same as the nth root of a times the nth root of b. n√(a/b)= (n√a)/(n√b) nth root of a over b is the same as nth root of a over nth root of b

Using a quadratic function to represent area with a fixed perimeter

x= width y = area P = perimeter Define length in terms of x: Length Is (P/2-x) Then. Multiply the length by the width: y=x(P/2-x)

Factoring quadratics where a=1

x^2+8x+12 Find two numbers that multiply to make twelve and add to make 8 (2 and 6) then turn the expression into this (x+2)(x+6) x^2+bx+c pq=c p+q=b (x+p)(x+q)

Finding vertex form from standard form by completing the square

y=2x^2+8x+5 take 5 to the other side y-5=2x^2+8x Factor out two y-5=2(x^2+4x) Complete the square by filling in the missing c value. y-1=2(x^2+4x+4) Factor the square y-1=2(x+2)^2 Add the one back in y=2(x+2)^2+1 Basically, take c to the other side, factor out a, and then add what would make a perfect square into that side of the expression (remember that your actually adding a times the value you put in to the other side). Then represent that as two sides of a square by factoring it. Add the value you have on the other side back in.

Simplifying negative square roots

√-6 √-1 * 6 i*√6

Multiplying complex numbers (Complex scale factor)

(a+bi)(c+di) a(c+di)+bi(c+di) ac+adi+cbi-bd ac-bd+i(ad+cb) On the complex plane it finds the product of bi and c+di and adds that to the product of a and(c+di)

Exponent rules (LOOK UP ON KHAN ACADEMY)

1.a^m*a^n=a^(m+n) You can add exponents if you are multiplying two with the same base 2.(a^m)^n=a^(mn) You can multiply exponents if you put an exponent to a power 3.(ab)^n=a^n*b^n You can distribute exponents. 4. (a/b)^n=(a^n)/(b^n) If a fraction being put to a power you can distribute the power to the top and bottom of the fraction. 5. a^m/a^n=a^(m-n) Dividing exponents with the same base is the same as subtracting the exponents. 6. a^(-n)=1/(a^n) If an exponent is negative you can find the reciprocal of that exponent and then make the exponent positive.

Factoring quadratics where a does not equal one using factoring by grouping

2x^2+8x+6 Find two numbers that multiply to equal 2*6 (ac) and add to equal 8 (6 and 2) 2x^2+6x+2x+6 Split the b into both of these terms 2x(x+3)+2(x+3) factor out the greatest common factor of both of the binomials (always try to make the x's positive by factoring out negative factors when needed) (2x+2)(x+3) Factor out the common binomial factor. 2(x+1)(x+3) Factor out any coefficients of x in the binomials. (Look up on Khan Academy if confused)

Multiplying complex numbers (imaginary scale factor)

3i(2+5i)=6i-15=-15+6i This rotates the vector 90 degrees counter clockwise and scales it by three. -3i(2+5i)=-6i+15=15-6i This rotates the vector 90 degrees clockwise and scales it by three

Quadratic function

A function that has an x squared term and no higher degree than x squared. y=ax^2+bx+c (standard form of quadratic equation) This function has a linear first order difference and a constant second order difference. The basic rate of change (if a is equal to 1) is always going to be, out 1, up 1, out 1, up 3, out 1, up 5, out one, up 7. Now referred to as {1, 3, 5, 7...} It is graphed with a parabola.

Perfect square trinomials

A trinomial that has been made by multiplying two of the same binomials together. For example, (x+3)(x+3)= x^2+6x+9 if ax^2+bx+c is a perfect square then c=(b/a2)^2 To find a missing c value factor out a (if a≠1) and then add in (b/2)^2 For example, 4x^2+8x+? 4(x^2+2x+?) 4(x^2+2x+1) 4x^2+8x+4 To find a missing b value factor out a and find square root of c times two and replace the missing b with that

Average of two complex numbers

Add the two complex numbers then divide that by two. Represented in the complex plane as the midpoint of the vector that connects them.

Solving for x intercepts in factored form

BEFORE FINDING X INTERCEPTS FACTOR OUT ANY COEFFICIENTS OF X IN THE BINOMIALS. if y=a(x+p)(x+q) then x intercepts equal (-p,0) and (-q,0) because 0=(x+p)(x+q) 0=x+p OR 0=x+q -q=x OR -p=x If confused look up zero product property on khan academy.

Simplifying radicals

Factor out the inside of the radical and then find something that you can take out from under the radical. Do this until you are sure that there are no more possible factors that could be taken out. If you take variables outside make sure to put absolute values around them. Look on Khan Academy for examples

Finding the vertex in factored form

Find line of symmetry using method mentioned above, then plug in that value for x and solve for y. This should give you the x and y value of the vertex.

Finding line of symmetry

Find the average of the x intercepts, this should be the x value of the line

Recursive equation for quadratics

Find the equation for the first order difference: Find the slope and then find what you have to add to the slope times the current x value (NOT f(x-1)) to get the increase from f(x-1) to f(x) this is the y intercept, use these values to find an equation for the first order difference. Then write a recursive equation with adding the explicit equation to f(x-1). F(x)=f(x-1)+mx+b

Finding factored form from a parabola

Find the stretch and the x intercepts. If stretch = 6 and x intercepts are (5,0) and (-8,0) then factored form is y=6(x-5)(x+8)

Solving Quadratic inequalities by graphing

Find the x intercepts and then give a rough sketch of the graph. Then give the areas where the parabola meets the requirements with set notation or interval notation. You may alter the solution to make sense if it is in a specific situation. For example, if the parabola needed to be less than 0 for this image then you would write {x | -1<x<2} or (-1, 2). If the parabola needed to be larger than or equal to 0 then you would write, {x| x≤-1 OR x≥2} or (-∞, -1] U [2, ∞)

Solving for a system of equations on the calculator

Go to the math menu on the graph and select Intersection. Choose the two lines that you want to solve for the point of intersection. When it asks for lower bound go on the left side of the point, and when it asks for the upper bound go to the right side of the point.

Solving for x intercepts with vertex form

How to solve for x intercepts from vertex form: y=2(x+1)^2+3, Plugin 0 0=2(x+1)^2-3 subtract three 3=2(x+1)^2 Divide by two (a) 3/2=(x+1)^2 Find square root of both sides ±√3/2=x+1 (DO NOT FORGET ± ON SIDE OF THE EQUATION WITHOUT X) Subtract one -1±√3/2=x Give both solutions: -1+√3/2=x OR -1-√3/2 NOTE: If you need to find the square root of a neg. number then there are no intercepts.

Finding the rate of change in an exponential function where x does not increase by an increment of a whole number.

If at x=0 y=4 and at x=1 y=8 then at x=1/2 y=? Find the rate of change by making an equation where r = rate of change. We know that the common ratio is going to have to be squared to get to 8 (because there are to jumps in between x=0 and x=1) and multiplied by 4, the starting value. SO, 8=4r^2 2=r^2 ±√2=r ±√2 is the common ratio for when x increases by one half each time. Therefore we multiply it by four and get ±√2*4 So y=±√2*4 at x=1/2 If we know that the equation for this function is y=4(2^x) So you could also just solve for 1/2 at x, y=4(x^(1/2))

Solving for a (the stretch) with two x intercepts and a point

If the x intercepts are (4,0) and (6,0) and the point is(0,3) then we know that the factored form of the equation is going to look something like this, y=a(x-4)(x-6) We know from the point that when y =3, x = 0. So we plug in those values and solve for a. 3=a(0-4)(0-6) 3=a(-4)(-6) 3=a(24) 3/24=a The stretch is 3/24. This is useful for when you're given two x intercepts and a point and then are asked to give something that requires the quadratic function(e.g. a missing value, the vertex, the function itself etc.)

Dividing two complex numbers

If you are dividing two complex numbers then multiply both the numerator and the denominator by the denominator's conjugate. This will give you a real number on the denominator with which to divide the numerator by. Example: __3+5i__ 5+1i __3+5i__ _*5-i_ = _20+22i_ 5+1i *5-i = 26 _10+11i_ 13 _10_ + _11i_ 13 13 Always remember to split the fraction up into the real part and imaginary part. It should be in the form a+bi

Tracking acceleration from gravity: Instantaneous speed

Instantaneous speed is tracked by multiplying the rate of acceleration by the time. For example, if the rate of acceleration was 12 feet/ second per second (or sec^2) the equation would v(t)=12t

Determining the nature of x intercepts with determinant

Plug in values for b^2-4ac, if it equals 0 then it has one real root, if it is a positive perfect square then it is rational and has two roots, if it is a positive non-perfect square then it has two irrational roots, if it is negative then it has two NON-REAL roots.

Solving quadratic inequalities using sign tables

Put it in factored form and then make a number line with the x intercepts on it. Put the different factors below the number line and put a sign below the line that says whether the factor is positive or negative at that point. Then multiply the signs and put that below to find whether the parabola is positive or negative on that portion of the line. This is basically like sketching out a graph but it tells you when the parabola is positive or negative. Example: Factored form of quad inequality (x+5)(x-3)≤0 ______________________________ -5 3 (x+5) - + + (x-3) - - + (x-3)*(x+5) + - + Answer: {x | -5≤x≤3} or [-5,3]

Complex number

Real Numbers, Non-real numbers(Purely imaginary numbers or imaginary numbers with a real component) can be represented as a+bi, where a is real part and bi is the imaginary part.

Tracking acceleration from gravity: Height

Subtract 6t^2 (formula for total distance traveled) from the height (let's pretend the height is 500 ft.) to get the height formula. h(t)=500-6t^2

Finding square root of i

The proof is in math binder, remember to rationalize denominators.

Finding two linear functions that when multiplied will make a parabola

The two linear functions must have the same x intercepts as the parabola. Their slopes must also multiply to equal the parabola's stretch. Keep in mind that the x intercept form of a line is m(x-c) where c is the x intercept. For example if you had a quadratic equation that had a stretch of 6 and x intercepts of 5 and -4 then the lines 3(x-5) and 2(x+4) would multiply to equal the parabola. The y intercept of the parabola will be the two y intercepts of the lines multiplied by each other

Modulus of complex number

This finds the distance of the number from the origin. For example 4+6i Represented as |4+6i| Use pythagorean theorem: √(4^2 +6^2) √16+36 √52 2√13

Tracking acceleration from gravity: average speed

To find the average speed from one point to another you add up the instantaneous speed at those points and then divide it by two. For example, if the inst. speed at 6 seconds was 5 feet per second, and the inst. speed at 8 seconds was 9 feet per second then you add 5 and 9 to get fourteen and the divide 14 by two to get 7 feet per second. To track the average speed of an object over the total time it has been falling you take half of rate of acceleration and then multiply that by the time. For example, if the rate of acceleration is 12 feet/ second per second you would represent it with the function, Vave(t)=12t/2 or Vave(t)=6t

Tracking acceleration from gravity: Distance traveled

To get distance traveled from one point in time to another you find the average speed between those points in time and then multiply it by the seconds in between those points. If you want to get the total distance an object has traveled while falling multiply the time by the current average speed. For example, if the rate of acceleration is 12 feet/second^2 use the current average speed Vave(t)=6t and multiply that by the current time t to get the distance. d(t)=6t^2

Proving Quadratic formula: Completing the square

Turn the algebraic form into vertex form by completing the square and then solve for x when y=0. This should get you the formula. MAKE SURE TO PRACTICE THIS.

Finding an explicit equation of a quadratic function (area model)

Turn the area model into a shape you can find the area of, then define that area in terms of x

How to solve for a (the stretch) when given the vertex and a point.

We are given the vertex of a quadratic function, (6,7) and then are given a point on the parabola, (8,4). We know that the vertex form of the equation will look like this, y=a(x-6)^2+7. We must find a. We know that when, x=4 y=8. Therefore we can plug those numbers into the equation, 4=a(8-6)^2+7 Simplify, 4=a2^2+7 4=a4+7 -3=4a -3/4=a -3/4 is the stretch. This is useful for the same reasons mentioned above.

Vertex form

What vertex form tells me about the parabola: y=3(x-5)^2+8 The 3 tells me that the change is multiplied by three (instead of {1, 3, 5, 7...} It is now, {3, 9, 15, 21...}). The parabola should be narrower. -5 tells me that the vertex's x value should be 5 (the opposite of negative 5). 8 tells me that the vertex's y value should be 8. How to find vertex form from a parabola: The vertex is at (-1, -4), the change is {1, 3, 5, 7...} so a has to be one. I can write y=(x+1)^2+4

Quadratic formula

You can use this formula to solve for the x intercepts on a quadratic equation. Get the quadratic equation into standard form 0=ax^2+bx+c then plug in a, b, and c into the formula. If you need to get the root of a negative number then there are no x intercepts. Mnemonic: There was a negative boy (-b) who couldn't decide whether or not (±) to go to a RADICAL party (√) but he was a square, (b^2) so he missed out on four awesome chicks, (-4ac) and it was all over at two am (/2a).