IB Mathematics 3.1

Sigma notation

addition of all terms on a sequence (right) starting (bottom) ending (top)

Pi notation

multiplication of all terms on a sequence (right) starting (bottom) ending (top)

Even functions

function y=f(x) in which f(-x) = f(x). In other words, the left side of the function graphed is a mirror of the right side.

Odd functions

function y=f(x) in which f(-x) = -f(x). In other words, the left side of the function graphed is a mirror of the right side reflected across the x-axis.

Result of graphing y = |f(x)| from y=f(x)

all points below the x axis will be reflected up

Result of graphing y = f(|x|) from y=f(x)

all points left of the y axis will be replaced by a reflection of the right side of the graph.

Concave up

Has a local minimum, end behavior y → infinity

Concave down

Has a local maximun, end behavior y → negative infinity

Vertical asymptote

a vertical line on a graph that a function can’t cross, mimicking it as y → infinity. Algebraically caused by dividing by zero and represented by the roots of the denominator.

Horizontal asymptote

a horizontal line on a graph that a function can cross, mimicking it as x → infinity. Algebraically caused by exponential growth and represented by the division of the leading coefficients of the numerator and denominator.

Holes in rational functions

caused by a common factor between the numerator and denominator, which can be factored out but will remain outside the function’s domain.

Denominators with imaginary roots in rational functions

when the denominator has no real roots (i.e. x²+1) there is no vertical asymptote.

Oblique asymptotes in rational functions

happens when the numerator is of a higher power than the denominator. Equation of the asymptote is the quotient of the rational function (can use synthetic division)

Reciprocal functions

for y= f(x), modeling y= 1/f(x) results in:

• Roots become vertical asymptotes

• All y values becoming their reciprocals

• Vertical asymptotes becoming roots with a hole

• Values at y=1 remain the same