**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} = r^{2}

- § 2:
- Lines and their slopes

slope-intercept: y = mx+b

point-slope: [(y-y_{0})/(x-x_{0})] = m

two-point: [(y-y_{0})/(x-x_{0})] = [(y_{1}-y_{0})/(x_{1}-x_{0})]

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=*a*f(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_{n}x^{n}+a_{n-1}x^{n-1}+¼+a_{1}x+a_{0} = 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) = ax^{2}+bx+c ; graph = *parabola*

Standard form: ax^{2}+bx+c = a(x-h)^{2}+k

complete the square: ax^{2}+bx+c = a(x^{2}+[b/a]x)+c

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

(and subtract corresponding amount outside!)

standard form ®graph:

x^{2} 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

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