College Algebra Acosta/Karwoski Slide 2 CHAPTER 1 linear
equations/functions Slide 3 CHAPTER 1 Sections 1 and 2 Slide 4
PRELIMINARY VOCABLULARY Slide 5 Vocabulary Expression vs equation
equivalent form Evaluate, simplify, solve, factor Information vs
question Process vs answer number, line, point, interval Slide 6
Linear equations Definition : Equations that have ONLY
multiplication and addition with degree 1 variables and terms.
Slide 7 Linear equations Slide 8 Change the form of a linear
equation Solve linear equations Finding points general, specific,
intercepts Slope of line/ rate of change Significance of
information: ie word problems Linear graphs Slide 9 SOLVING LINEAR
EQUATIONS General solutions, intercepts Slide 10 Solving- finding a
value for the variable(s) Algebraically One variable: Two variable
specific general intercepts Forms of equations y = mx + b Ax +By +
C = 0 (y y 1 ) = m(x x 1 ) From graph- entering equations window
settingw [xmin, xmax,xscale] x [ymin, ymax, yscale] From table
Slide 11 Solving algebraically for one variable There are 2 numbers
in an equation that are equal Changing one of them changes the
truth of the equation changing both of them the same way changes
the form of the equation but not the truth of the equation Isolate
the variable changing the equation so that the variable is on one
side (alone), makes a statement about the value of x that is true.
Slide 12 Examples: solving for one variable Slide 13 Solving for 2
variables 2 variables in an equation form a relation between the
two variables A solution to the equation is a pair of numbers For
any 2 variable equation there are infinite solutions Slide 14
Specific points x48 y-312 Slide 15 General solutions For 3x + 2y =
9 find 5 solutions For y = 3x/5 2 find 3 solutions Slide 16
Intercepts An intercept is a point- a solution pair where a graph
crosses the x- axis or the y- axis. ( x-intercepts are frequently
called roots ) An x-intercept (root) has a y-coordinate of zero A
y-intercept has an x- coordinate of zero Thus intercepts are also
sometimes referred to as the zeroes of the equation Slide 17
Finding intercepts : algebraically.2x -.6y = 2.4 y = 3x 7 3000 + y
= 6x Slide 18 Using a graph to check solutions Setting the graphing
window Entering a graph Using the trace key Slide 19 Examples:
check solutions Slide 20 Using a table Enter equation 3(x 9) + 5x 4
= 15 Table set Look up Guess and check Slide 21 SLOPE AND RATE OF
CHANGE Slide 22 Slope/ rate of change Slope is a measure of the
angle of a line Rate of change is a ratio used to compare the
change in two numeric items. Slope IS a rate of change for a line
it is constant Slide 23 What you should know about slope the line
is increasing if and only if the slope is positive The line is
decreasing if and only if the slope is negative If the line is
horizontal the slope is 0. If the line is vertical the slope is
undefined Slope is a ratio between changes in two different things
- in other words slope is a rate rates MULTIPLY so in y = mx + b m
is the slope of the corresponding linear graph Slide 24 slope Given
points Given graph Given equation Slide 25 Examples find the slope
of lines that pass through the following points (2,9) (-3,8) (-6,3)
(3, - 4) ( 0, 6) (5,6) (-5, 4) (-5,8) Slide 26 More examples Slide
27 Rate of change/ slope Slide 28 Examples: find the slope of each
line described here Slide 29 Sketching linear graphs Slide 30
APPLICATIONS Significance of information Slide 31 Understanding
answers in a real life graph x and y have labels Therefore ordered
pairs have labels The label on slope comes from these labels Ex: a
graph represents the calories consumed by running over a period of
hours. therefore y is calories x is hours The point (2, 4034)
means? The point (0,500) would mean? Is this point likely to be a
solution point? What about (-3, 268)? What does (4, 0) mean? If the
graph is linear and its slope is 5297/2 what does this number mean?
Slide 32 Example y =.3x + 32 is the equation for your bill in $
under a phone plan where x is the number of minutes you talked. If
you talk for 20 minutes what is your bill? If your bill is $49.10
how many minutes did you talk? What is the y-intercept - what does
it mean? What is the slope? What does it mean? What is the x
intercept ? What does it mean? What is an appropriate domain
(window) for this problem? Slide 33 CHAPTER 1 SECTION 3 Functions
vs relations Slide 34 Relation A relation is any statement, table,
graph, diagram that creates a connection between two sets. the sets
are called domain and range any equation that contains 2 variables
is a relation Functions are not always straight lines Functions are
not always equations Slide 35 Examples: these are all relations y =
3x + 5 x 2 + y 2 = 12 x224-65.16 y-423786 Slide 36 Domain and range
Sets of numbers Domain the set of numbers that are reasonable
replacements for the x coordinate Range the set of numbers that
correspond to the numbers in the domain thus forming solution
pairs. Domain and range are NOT single numbers. The are frequently
interval of numbers Slide 37 Examine the domain and range y = 3x +
5 x 2 + y 2 = 12 x224-65.16 y-423786 Slide 38 Restricted domains
Domains can be restricted by choice (as in choosing a window for
the calculator) Domains can be restricted words in an application
problem Domains can be restricted by operators The natural domain
of a function is all real numbers unless some numbers are
restricted by one of the above, in which case the domain is all
numbers except those restricted values Domain is an interval Slide
39 Restrictions you should know Fractions cannot have a zero in the
denominator - therefore any values that create a zero in the
denominator are not in the domain square roots (4 th roots, 6 th
roots, etc) cannot have a negative number inside the radical sign
therefore any values that cause the radicand to be negative are
restricted Slide 40 Examples Slide 41 Functions A relation with the
characteristic that each element in the domain is related to ONLY
one value in the range is a function Domain takes on the sense of
an input value and range is the output. The relation now is about
how the value of (range variable) DEPENDS on the value of (domain
variable) Which set is chosen for domain affects whether you have a
function or not. Slide 42 Example: decide if the following
relations are function a report is created that connects gpa to
student number : domain ? range? function? Slide 43 Examples
continued: is the relation a function 2x + 2y = 14 y = 3x 2 5x
NOTE: Any equation that can be solved uniquely for y is a function
of x. X123456789 y642101234 Slide 44 Function notation The symbols
g(x) = y mean: There is a rule (not necessarily an equation) called
g that has an input of x with a result of y What does h(2) = -4
mean? therefore k(x) = 2x + 5 means? Slide 45 Questions using f(x)
notation given what is f(2014)? for what value does k(x) = 92? what
is the domain of the function? what is the range of the function?
X2012201420162020 K(x)659243120 Slide 46 Questions using f(x)
notation Slide 47 Using table feature on calculator X 64.832.41
n(x) Slide 48 CHAPTER 1 SECTION 4 Interpreting/reading function
graphs Slide 49 Information from graphs - for ANY relation not just
linear Do I have a function? Domain - (the interval covered by the
graph left to right) Range (the interval covered by the graph down
to up) Given input find output ( f(x) = ? - find a point) Given
output find input (f(??) = y - find a point Find y- intercepts (a
point on y axis) Find x intercepts (a point on x axis) Slide 50 Is
it a function? If output is unique for input yes Vertical line test
Slide 51 Domain and range Stated as intervals. Pay attention to
closed and open endpoints and arrows. Note any skipped sections
Slide 52 Given input estimate output f(2) = 3 means that (2,3) is a
point on the graph. find f(5) means to find the output when the
input is 5. If the graph is a function output is unique but not
always defined. Find: k(2) k(-3) k(0) k(4) k(x) Slide 53 Given
output estimate input (solve) Find a value for x so that f(x) = 9.
means find the input value that gives you 9 There can be MORE than
one answer to this question. There can be no solution to this
question!!!!! Find x so that: m(x) = 5 m(x) = 2 m(x) = 0 m(x) = -8
Slide 54 Y and x intercepts solution points on the axis a function
has at MOST one y intercept but may have 1 x-intercept, no x
intercepts or many x- intercepts Slide 55 New questions using
graphs Where is the function Increasing (intervals) Where is the
function Decreasing (intervals) On what intervals is f(x) < # On
what intervals is f(x) > # Slide 56 Increasing and decreasing
intervals This question is about the domain. Increasing is going up
when read from left to right Decreasing is going down when read
from left to right Slide 57 f(x) > a f(x) Example Estimate the
values for which f(x) < 2 f(x) > 7 f(x) 0 Slide 59 CHAPTER 1
SECTION 5 Linear functions Slide 60 Linear functions are linear
equations in 2 variables Linear equation in 2 variables Contains
only multiply and add can be written as y = mx +b as long as the
equation contains a y variable Always graph a line rate in equation
is same as slope of line Given y you can find x Given x you can
find y Linear function Has only multiplying and adding ALL linear
equations are functions EXCEPT x = constant Is the only function
that graphs a straight line Has a constant rate of change ( either
increases everywhere or decreases everywhere) Has one y intercept
Has at most one x intercept domain and range are both ALL real
numbers - f(x) = y Slide 61 Models - equations A model is a
mathematical equation that describes the data for a real life event
where there is a pattern that matches that equation Some models are
exact like interest/time models Some models are estimates like
distance/time If there the pattern in the data has a constant rate
of change it is linear. Slide 62 Finding equations for linear
models a line is described by 2 points It can also be described by
one point and its slope If you can describe or picture the line you
can determine the equation that generates the line This can be done
by many different methods - We will explore 3. Slide 63 y = mx +b
method Given the line through (0,2) with slope of 3/5 Given the
line through (2,7) with slope of 5 Given the line through (-4,7)
and (5,3) Slide 64 Point slope method Slide 65 Parallel and
perpendicular lines parallel lines have the same slope Write an
equation for a line parallel to y= 3x 9 going through the point
(2,7) Slide 66 Horizontal and vertical lines A horizontal line has
slope of 0. A horizontal line has a y intercept --- (0,b) and ALL
other points have a y co- ordinate of b So the equation will be y =
b A horizontal line through (9, 7) has an equation of? A vertical
line has NO slope is NOT a function therefore does NOT contain a y
variable - and has an x intercepts or (x,0) where ALL points have
the same x value. So the equation for a vertical line through (9,7)
is?
