IB Math SL 1 Midterm

Complex conjugates

(a+bi)(a-bi)=a^2 + b^2 (a-bi)(a+bi)=a^2 + b^2

Sin^2t+cos^2t=

1

Graphing a quadratic function in the form f(x)=ax^2 +bx+c

1) Determine how the parabola opens (if leading coefficient is -, opens down, if + opens up) 2) find the vertex using x=-b/2a and then plugging that value in for y 3) aos is x coordinate of vertex 4) Find the x intercepts by solving f(x)=0 5) find y intercept by finding f(0) 6) graph

Transformation of a f(x)=x² graph, such as f(x)=-2(x+1)²+1

1) Make a parent table x y -2 4 -1 1 0 0 1 1 2 4 2) Subtract 1 from x values 3) Multiply y values by -2 and add 1

Finding the inverse of a function

1) Replace f(x) with y in the equation for f(x) 2) Interchange x and y 3) Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function. 4) If x has an inverse function, replace y in step 3 by f⁻¹(x). We can verify our result by showing that f(f⁻¹(x))=x and f⁻¹(f(x))=x

Synthetic division by x-c

1. Arrange the terms in descending order, filling in zeros for missing values 2. Put c as the divisor(opposite of the number in x-c) 3. Multiply, then add until no more columns open 4. Final number is the remainder

Long Division of polynomials

1. Arrange the terms of both the dividend and the divisor in descending powers of the variable. 2. Divide the first term in the dividend by the first term in the divisor. 3. Multiply every term in the divisoe by the first term in the quotient. Subtract the product from the dividend, and bring down the next term. 4. Write the remainder over the divisor

Graphing rational functions

1. Determine if there will be symmetry f(-x)=f(x) y axis symmetry f(-x)=-f(x) origin symmetry 2. Find y intercepts by setting x equal to zero 3. Find x intercepts by setting y equal to zero 4. Find asymptotes 5. Plot extra points

How to find the minimum and maximum of quadratic functions in the form f(x)=ax^2 +bx+c

1. If a>0, then f has a minimum that occurs at x=-b/2a. This minimum value is f(-b/2a). 1. If a<0, then f has a maximum that occurs at x=-b/2a. This maximum value is f(-b/2a). In both cases, the value of x gives the location of the max/min, but the y value actually is the max/min

Graphing logs

1. Put it into exponential form so that its number^y=x and pick y values not x 2. Do transformations 3. It will have a vertical asymptote 4: range is all real numbers

Finding zeros of polynomial functions and their multiplicities

1. Set the polynomial function equal to zero, and factor it 2. The exponent of the factors is the multiplicity 3. If the multiplicity is even, the graph touches the x axis and if it is odd, the graph crosses the x axis

Compound interest

A=P(1+(r/n))^nt P:principal, what you start with R: rate as a decimal N: number of times per year (ie if it is monthly, n=12) T: amount of time in years

Continuous compounding formula

A=P*e^rt

Graph of y=Asin(Bx-C)

Amplitude: absolute value of a Period: 2pi/b Phase shift: c/b

General form

Ax+By+C=0

Even and odd trig

Cos and sec are even so - sign doesnt go to front

1+cot^2t=

Csc^2t

F(x)=b^x basic characteristics

It is exponential D is all read numbers Has a horizontal asymptote

Logb(MN)

LogbM+logbN

Logb(M/N)

LogbM-logbN

F(x)=e^x basic characteristics

Looks like exponential Same as exponential in terms of graphing

LogbM^P

PlogbM

1+tan^2t=

Sec^2t

Vertical asymptotes

Set the denominator equal to zero, but see if you can cancel anything out first

Domain of logs

Take the stuff in parentheses like in y=log4(x+3) Make x+3>0 and that also gives u your vertical asymptote

Even vs odd function

The function f is an even function if f(-x)=f(x) Basically, the equation does not change if x is substituted with -x The function f is an odd function if f(-x)=-f(x) Basically, every term on the right side of the equation changes sides if x is substituted with -x

Radian measure

Theta=s/r S is arc length R is radius Angle in radians!!!

Linear speed

V=rw

Linear speed

V=s/t Time is t S is arc length

Angular speed

W=theta/t

Point slope form

y-y₁=m(x-x₁)

Horizontal asymptote

• if the degree of the numerator is less than the degree of the denominator, the asymptote is y=0 •if n=m, then divide the leading coefficient of the numerator by the leading coefficiant of the denominator. Your answer, set equal to y, is the horizontal asymptote • if n>m, there is a slant asymptote