Transcript y=f(x)

STUDY GUIDE FOR TEST 2
Name:__________________
Project 3: Fill in the blanks and do the assigned questions.
11/6/07
Quadrant I
Quadrant II
(
ORDERED PAIR:
The first number in the ordered pair
is the __-coordinate and the second
number in the ordered pair is the
__-coordinate.
, )
__-axis
Origin
( , )
Quadrant III
Linear Equations
__-axis
Slope = change in __
change in __
Lines l1 and l2 are
Slope of l1 =
___________ to each other.
The slope of l1 is the ______
as the slope of l2.
l3
The slope of l1 is the
_________
l1
x-intercept of l1 is ( , )
To find the x-intercept,
set__ =0 and solve for x.
-
The slope-intercept form of a linear equation
is y = ______________, where m is the slope
and ( 0 ,b ) is the y-intercept.
The point slope formula for a line with point
(x1, y1) and slope m is
___________ = __ (_____________)
Equation for l2 is
y =____________
l2
Equation for l3 is
y =____________
_________ of l3.
Given two points on a line
Slope of l3 =
(x1,y1) and (x2,y2)
The slope can be found through the equation
m=
Equation for l1 is
y= ___________
y-intercept of l1 is
( , ) To find the
y-intercept, set __
2
=0 and solve for y
1
-2 -1
1 2
Lines l1 and l2 are
___________ to l3.
Quadrant IV
Slope of l2 =
2
1
-2 -1 -11 2
-2
Equation for this
horizontal line is
_____________
This line has a slope of __
Equation for this
vertical line is
_____________
This line has NO SLOPE
LINEAR INEQUALITIES
To graph a LINEAR INEQUALITY,
First rewrite the inequality to solve for y.
If the resulting inequality is y > ….,
Then make a dashed line and shade the area ____ the line.
If the resulting inequality is y < …..,
Then make a dashed line and shade the area _____ the line.
If the resulting inequality is y≥ …..,
Then make a _________ line and shade the area ____ the line.
If the resulting inequality is y ≤ …….,
Then make a ________ line and shade the area _____ the line.
Graph the solution set of 3x – 2y ≤ 12
FUNCTIONS
Input x
Function
f(x)
Input 3
Output
Function
f(x)=2x-1
y=f(x)
Output
f(3) =
__________
A function, f, is like a machine that receives as input a number, x, from the domain,
manipulates it, and outputs the value, y.
The function is simply the process that x goes through to become y. This “machine” has 2
restrictions:
1.
It only accepts numbers from the domain of the function.
2.
For each input, there is exactly one output (which may be repeated for different inputs).
“OFFICIAL” DEFINITION OF A FUNCTION:
Let x and Y be two nonempty sets. A function from x into Y is a relation that
associates with each element of X, exactly ___ element of Y.
However, an element of Y may have more than one elements of x associated with it.
That is, for each ordered pair (x,y), there is exactly __ y value for each x, but
there may be multiple __ -values for each y. The variable x is called the
independent variable (also sometimes called the argument of the function), and the
variable y is called dependent variable (also sometimes called the image of the
function.)
Analogy: In the x-y “relation”-ship, the x’s are the wives and the y’s are the
husbands. A husband is allowed to have more than one wife, but each wife(x) is only
allowed 1 husband(y).
A relation is a correspondence between two sets. If x
and y are two elements in these sets and if a relation
exists between x and y, then x corresponds to y, or y
depends on x.
Hours Studying in Math Lab
Score on Math Test
x
y
Hours Studying in
Math Lab
Score on Math Test
2
3
4
4
5
6
6
7
60 70 70 80 85 85 95 90
The set of x-coordinates {2,3,4,4,5,6,6,7} corresponds to
the set of y coordinates {60,70,70,80,85,85,95,90}
100
The set of distinct x-coordinates is called the _______ of
the relation. This is the set of all possible x values
specified for a given relation.
The set of all distinct y values corresponding to the xcoordinates is called the __________.
In the example above,
Domain = {2,3,4,5,6,7} Range = {60,70,80,85,85,95,90}
80
60
40
20
0
0
2
4
6
Score on Math Test
8
This relation is not a function because there are
two different y-coordinates for the x-coordinate,
4, and also for the x-coordinate, 6.
WORD PROBLEMS
Simple Interest
Interest earned ($$) = Principal * Rate * time
(initial investment) * (interest rate )* (1 year for annual interest)
Mixture Problems
Quantity of a Substance in a solution = % of concentration * Amount of Solution
Example: Forty ounces of a 60% gold alloy means that the quantity of gold in the alloy
is .60 * 40 = 24 ounces of gold.
Distance Problems
Distance = rate * time
If distance is in miles, and rate is in mph, then time must be in _________
If time is in minutes, then multiply time by --- to convert to hours.
Setting up word problems:
1) Find out what you are being asked to find. Set a variable to this unknown
quantity. Make sure you know the units of this unknown (miles?, hours? ounces?)
2) If there is another unknown quantity, use the given information to put that
unknown quantity in terms of the variable you have chosen.
(For example, if total distance traveled is 700 miles, then part of the trip is x miles
and the other part of the trip is 700 – x miles.)
3) Set up a table with a row for each unknown and columns made up of the terms of
one of equations above (r*t = d, Pr = I, etc..)
4) Use the given information to combine the equations of each row of the table into
one equation with one variable to solve for.
5) Once one variable is solved for, you can find the other unknown. (For example, is
x = 100 miles, then the other part of the trip is 700 – 100 = 600 miles.)
6) Check your equation by plugging in your value for x and seeing if your equation
is true.
SYSTEMS OF LINEAR EQUATIONS
A system of linear equations is a set of two equations of lines.
A solution of a system of linear equations is the set of ordered pairs that makes each equation
true. That is, the set of ordered pairs where the two lines intersect.
If the system is _______________, there is ONE SOLUTION, an ordered pair (x,y)
If the system is ______________, there is INFINITELY MANY SOLUTIONS, all (x,y)’s that
make either equation true (since both equations are essentially the same in this case.
If the system is _______________, there are NO SOLUTIONS, because the two equations
represent parallel lines, which never intersect.
GRAPHING METHOD.
Graph each line. This is easily done be putting them in slope-intercept form, y = mx + b.
The solution is the point where the two lines intersect.
SUBSTITUTION METHOD.
2x – y = 5
3x + y = 5
Choose equation to isolate a variable to solve for. In this system, solving for y in the second
equation makes the most sense, since y is already positive and has a coefficient of 1.
This second equation turns into y = -3x + 5
Now that you have an equation for y in terms of x, substitute that equation for y in the first
equation in your system.
Solve for y by substituting x=2 into y = -3x + 5.
Substitute y = -3x + 5 in 2x – y =5
y = -3(2) + 5 = -6 + 5 = -1
2x – (-3x + 5) = 5
Therefore solution is (2,-1)
Simplify and solve for x.
CHECK by substituting solution into the other equation and
2x + 3x – 5 = 5
see if it is true.
5x – 5 = 5
3x + y = 5
5x = 10
3(2) + (-1) =5
x=2
5 = 5 YES!
ADDITION METHOD
5x + 2y = -9
12x – 7y = 2
Eliminate one variable by finding the LCM of the coefficients, then multiply both sides of the
equations by whatever it takes to get the LCM in one equation and –LCM in the other
equation. After this we can add both equations together and eliminate a variable.
Let’s choose to eliminate y. The y terms are 2y and -7y. The LCM is 14.
Multiply the first equation by 7 and the second equation by 2.
7(5x + 2y) = -9(7)
Solve for y by substituting x=-1 into either of the two original
2(12x – 7y) = 2(2)
equations.
35x + 14y = -63
5(-1) + 2y = -9 2y = -4
y = -2
24x – 14y = 4
Therefore solution is (-1,-2)
Add them together
CHECK by substituting solution into the other equation and
59x = -59
see if it is true.
x = -1
12 x – 7y = 2 12(-1) – 7(-2) = -12+14 = 2 2 =2 YES!
REVIEW QUESTIONS
p.242-243 #7,11,13
p. 322-323 #1-25 odd
p. 365 #1-23 odd