Chapter 1: Functions and their graphs

§ 1:

Graphs of equations

graph = all points that satisfy the equation

How to graph?

plot points (and fill in gaps)

use x- and y-intercepts

use symmetry

y-axis: (a,b) on graph, so is (-a,b)

x-axis: (a,b) on graph, so is (a,-b)

origin: (a,b) on graph, so is (-a,-b)

Eqn for circle: (x-h)²+(y-k)² = r²

§ 2:

Lines and their slopes

slope-intercept: y = mx+b

point-slope: [(y-y₀)/(x-x₀)] = m

two-point: [(y-y₀)/(x-x₀)] = [(y₁-y₀)/(x₁-x₀)]

same slope: lines are parallel (do not meet)

lines are perpendicular: slopes are negative reciprocals

§ 3:

Functions

inputs = domain; outputs = range/image; f:A®B

y=f(x) : `y equals f of x' : y equals the value assigned to x by the function f

f,x,y, etc. are all placeholders; any other sybols are `just as good'

`implied' domain of f: all numbers for which f(x) makes sense

§ 4:

Graphs of functions

graph = all pairs (x,f(x)) where x is in the domain of f

all functions have graphs, but not all graphs `have' functions

function takes only one value at a point; vertical line test

symmetry (for functions)

y-axis: even function, f(-x) = f(x)

x-axis: XXXXXX

origin: odd function, f(-x) = -f(x)

increasing on an interval: if x > y , then f(x) > f(y)

decreasing on an interval: if x > y , then f(x) < f(y)

constant

§ 5:

Translations and combinations

shift to right by c; y=f(x-c)

shift to left by c; y=f(x+c)

shift down by c; y=f(x)-c

shift up by c; y=f(x)+c

y=af(x) ; stretch graph by factor of a

reflect graph along axes

y-axis: y=f(-x)

x-axis: y=-f(x)

combining functions: combine the outputs of two functions f,g

f+g, f-g, fg, f/g

composition: output of one function is input of the next

f followed by g = g°f; g°f(x) = g(f(x)) = g of f of x

§ 6:

Inverse functions

find a function g so that g(f(x)) = x for every x

magic: f undoes g ! Usual notation: g = f^-1

Problem: not every function has an inverse.

need g to be a function; so f cannot take the same value twice.

horizontal line test

Graph of inverse: if (a,b) on graph of f, then (b,a) is on graph of f^-1

graph of f^-1 is graph of f, reflected across line y=x

Chapter 2: Polynomials

§ 1:

Quadratic functions

polynomial = bunch of monomials = a_nxⁿ+a_n-1x^n-1+¼+a₁x+a₀ = f(x)

a_n ¹ 0, then n=degree of f

deg=0: constant fcn; deg=1: linear fcn; deg=2: quadratic fcn

f(x) = ax²+bx+c ; graph = parabola

Standard form: ax²+bx+c = a(x-h)²+k

complete the square: ax²+bx+c = a(x²+[b/a]x)+c

add half of [b/a], squared, inside parentheses

(and subtract corresponding amount outside!)

standard form ®graph:

x² to (x-h)² (shift left/right) to

a(x-h)² (stetch/reflect) to a(x-h)²+k (shift up/down)

lowest/highest point of graph = (h,k) = vertex of parabola

axis of symmetry: vertical line x=h

§ 2:

General properties of polynomials

graph has no gaps, hole, or jumps (f is continuous)

can draw graph without lifting up writing implement

graph has no corners - no sudden turns; graph is smooth

behavior at `ends':

n even, a_n > 0 : high/high

n even, a_n < 0 : low/low

n odd, a_n > 0 : low/high

n odd, a_n < 0 : high/low

root (zero) of f ; f(a) = 0 ; grpah of f hits x-axis at a

if f(a) = 0, then f(x) = (x-a)g(x)

nth degree polynomial can have at most n roots

nth degree polynomial can turn around at most (n-1) times

consequence of continuity: intermediate value theorem

if a polynomial takes on two values c and d, then

it also takes on every value in between

application: `finding' roots: if f(a) < 0 and f(b) > 0, then

there is a root of f somewhere between a and b