Chapter 1: Functions and their graphs
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
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
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
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
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
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
polynomial = bunch of monomials = anxn+an-1xn-1+¼+a1x+a0 = f(x)
an 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
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, an > 0 : high/high
n even, an < 0 : low/low
n odd, an > 0 : low/high
n odd, an < 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