Download Mod 1 Ch 3 - Linn-Benton Community College

Mth 95
Module 1
Spring 2014
Section 3.1 - Graphing Equations
The Rectangular Coordinate System, which is also called the _____________
____________________ system. On the grid below locate and label the
following: Origin (_______)
x-axis (_______) __________________
y-axis (_______) __________________
Quadrants I, II, III and IV
Points plotted in a plane are called _____________ ____________ because
the order in which they are written tells you which number is associated with
which axis. Plot each point below and tell the quadrant in which the point lies or
the axis on which it lies. In the first point (3,-2), 3 is called the
______________________and -2 is called the _________________________.
Point
Quadrant
or axis
(3,-2)
(0,3)
(-4,1)
(-1,0)
(-2 ½ ,-3)
Solutions of equations in two variables consist of two numbers that form a
true statement when substituted into the equation.
Determine whether each point is a solution of the equation.
y = -2x + 7
(1, 5)
(-2, 3)
Linear Equations: Standard form (in two variables)
The equation for a line is in standard form when it is written as
Ax + By = C, where A, B, and C are constants with A and B not both 0.
For each of the following examples identify A, B, and C.
5x – 2y = -1
Chapter 3
-2x – y = 9
2y = 5
x+y=3
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A graph is a picture of all the _____________________ of an equation.
To graph a linear equation we only need _______points.
Quick way to Graph a line in Ax + By = C form
– Find the intercepts of a line algebraically

To find the x-intercept of a line, let y = 0 in its equation and solve for x.

To find the y-intercept of a line, let x = 0 in its equation and solve for y.
Find the x and y intercepts for 4x – 3y = -12.
x-intercept 4x – 3(0) = -12
4x = -12
x = -3
y-intercept 4(0) – 3y = -12
-3y = -12
y=4
Use these intercepts to sketch a graph of the equation.
Be sure to use arrows.
Find the intercepts of 2x + 3y = 6.and then graph it.
x-intercept
y-intercept
Non-Linear Equations: When graphing non-linear equations more points will
be needed to create the graph.
On the graph of the quadratic equation: y  x 2  2 x  3 , each square is 2 units.
Locate and label: The vertex:___________
The Axis of Symmetry (AOS):__________
The x-intercept(s): ______________
The y-intercept: ______________
Note: for an equation with two variables you cannot list all the _______________
Chapter 3
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Spring 2014
Fill out the table and sketch the graph of y  x 1
x 1  y
x
0
1
2
3
4
5
The absolute value of a number is the distance of that number from zero.
Since distance is always measured in positive units, the absolute value of any
number will always be positive. The notation is: y  x . See Example 7 on page
124 of your test for the graph of this equation.
Graph the equation: y  x 1
x
-2
-1
0
1
2
x
-1
0
1
2
3
y
1
Graph the exponential function y   
2
x
Graph the equation: y  x 1
-2
y
Chapter 3
-1
0
1
2
y
x
Graph the quadratic function y = x2 – 1
x
-2
-1
0
1
2
y
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Mth 95
Module 1
Spring 2014
Section 3.2 - Introduction to Functions
Vocabulary of Relations and Functions
A _______________ is any set of ordered pairs, (x, y). The set of all first
components of the ordered pairs is called the _________________ or _____________ of
the relation. The set of all second components is called the _______________ or
____________ of the relation.
A _______________ is a relation in which each first component of the ordered pairs, the
____________, corresponds to exactly one second component, the _______________.
No two ordered pairs of a function can have the same first component and different
second components, that is, each input has a unique output.
Relations and functions can be displayed in six ways.
For each of the following, state the domain and range and tell whether it is a
function. For finite sets give the domain and range in set notation. For infinite
sets give the domain and range in interval notation.
1) as a set of ordered pairs
Domain:
 0,2, 1,4 , 1,2 ,  2,3 , 3, 2 
Range:
Function:
2) in a table
Input, x
0
1
2
3
4
5
Domain:
Output, y
-2
4
-1
-2
3
2
Range:
Function:
3) in a diagram For the relation in the diagram below, list the set of ordered pairs.
x
y
Domain:
1
0
2
-1
3
Range:
-2
Function:
Remember a function has only one output for each valid input.
4) function displayed in an equation
2
y   x  3  Domain ______________ Range ______________ Function _______
y  x4
Chapter 3
Domain _______________Range ______________ Function _______
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Review function notation:
f(x) is read “f of x”. It represents the function f written in terms of x.
y = f(x) so x is the ____________________ variable, the ________or _______
and f(x) is the __________________ variable, the ________ or ________
Note: f(x) does not mean f times x.
5) function displayed on a graph (See examples 5 and 6 on pages 132 – 134)
When a relation is graphed, we use the vertical line test to determine whether the relation is a
function.
VLT – If no vertical line can be drawn so that it intersects the
graph more than once, then the graph represents a function.
Do the following graphs display functions? Why or why not?
Each square represents 1 unit.
4 x 2  9 y 2  36
h( x )  x  4
Domain
____________________
____________________
Range
____________________
____________________
Function
________
_________
Identify each of the following points on the graph of h(x) above.
Point A
(-2, h(-2)) is the point _________ h(-2) =______
Point B
(-3, h(-3)) is the point _________ h(-3) = ______
Point C
(5, h(5)) is the point _________
h(5) = ______
Point D
(0, h(0)) is the point _________
h(0) = ______
For which x-values does h(x) = -3? ___________ or ____________
6) Functions expressed in words
Write the symbolic description for the area of a triangle whose height is 3 inches more
than its base, x. State the domain and range. (Remember the domain of a function is all
inputs that produce valid outputs.) Evaluate the function for x = 3 and interpret the
results.
Chapter 3
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In 2008, LBCC is paying $0.48 per mile when a faculty member uses his/her own vehicle
for transportation to a sanctioned event. Write a symbolic description for the cost, C(x),
of driving x miles. State the domain and range. Find C(130) and interpret what it means.
Write the symbolic description for the perimeter (p) of a square with side x. Evaluate for
an input of 7cm and describe what it means.
 2
Given the function: f ( x)  3 x  5 find f   
 3
Given the function: g ( x)  x 2  3
find
g (2) and g (a)
The function P(t ) 1.08(1.0139)t represents the population in billions in India where t is
the number of years after 2005. Find P(0) and P(12) and describe what they represent.
Section 3.3 Graphing Linear Functions
Review: Identifying linear functions represented symbolically
A function is linear if it can be written in slope-intercept form, f(x) = ax + b
(or y = mx + b). Which of the following are linear functions? Why or why not?
f(x) = 8 – 2x
Yes or No
____________________________________
f(x) = 5
Yes or No
____________________________________
f(x) = 2x2 - 1
Yes or No
____________________________________
Chapter 3
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4  3 x  2y
Yes or No
____________________________________
y  x2
Yes or No
____________________________________
x
x6
Yes or No
____________________________________
Yes or No
____________________________________
y
1
f  x  2 
3
x
Determining whether the function represented numerically in a table is linear
For each unit of increase in the input is there a constant change in the output?
x
-2 -1 0 1 2
f(x) 4 8 12 16 20
Constant change in f(x) = _________
Constant change in x = ________
Linear? _______
x
0 1 2 3
f(x) 1 4 9 16
Change in f(x) = _____________
Change in x
= ______
Linear? _______
Determining whether the relation represented in each graph is a linear function.
Which of the following graphs represent linear functions (caution!)?
Determine whether the following verbal descriptions are of linear functions. If they
are, define your variables and write an equation to represent it. Remember that linear
functions must increase or decrease at a constant rate.
The cost of a number (n) of first class stamps is proportional to the number of stamps you
purchase. One first class stamp costs $0.42.
The Maytag repairman comes to my house to repair my washing machine. He charges
me $40 per hour for the work he does and an additional $35 for the "house call".
Chapter 3
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Mth 95
Module 1
Spring 2014
A potent antibacterial medication is so effective it kills about half the bacteria in a wound
every day it is being applied.
Solving Applications
As a gas, like helium, is heated, it will expand. The formula: V(t) = 0.147t + 40
calculates the volume in cubic inches of a sample of helium at the temperature, t, in
degrees Centigrade (Celsius).
a) EvaluateV(0) and interpret what it means.
b) If the temperature increases by 10 degrees, how much does the volume of this sample
increase?
c) What is the volume of this sample of helium at 100 degrees?
d) At what temperature will this sample of helium reach a volume of 50 cubic inches?
Slope-intercept form of a line
The line with slope m and y-intercept (0, b) is given by
Remember the two easiest ways to Graph Lines –
Slope-intercept: Transform the equation to slope intercept form, plot the yintercept, use slope to plot other points on the line, and draw the line through the
points you’ve plotted..
X and Y intercepts: Replace x with 0 to find the y-intercept and y with 0 to find
the x-intercepts. Plot these points and draw a line through them.
y = 3x + 1
y = -2x -1
f(x) = 3 – x
Chapter 3
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Find the x and y-intercepts of 4 x  3 y  12 and use them to sketch the graph.
Other Methods to Graph Lines
Use a table of values to graph
the equation: 3 y2 x  6 . To make this
easier, solve for y and then choose values
for x that are multiples of 3 to avoid
fractions.
Graph these two equations on the
same set of axes. What do you
notice about the lines?
x   3 and y  2
Which graph is not a function?
Section 3.4 - Slope of a Line
The slope of a line passing through points (x1, y1) and (x2, y2) is
m=
Find the slope of a line that goes through (-3, 6) and (4,-2).
Slope can be found by counting the rise and run on a graph or using the coordinates of
two points in the above formula.
If the line rises from
left to right, the slope
is always__________
Chapter 3
If the line falls from
left to right, the slope
is always__________
Horizontal lines
always have
__________slope
Vertical lines
always have
___________ slope.
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Mth 95
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Spring 2014
Given a point on the line and slope, graph the line.
Steps:
(2, 1) m = ½
1) Plot the point given
2) Use the slope to find
and plot a second point
3) Draw the line
(-1, 0) m = -3
Given a graph of a linear equation, write the equation in slope-intercept form.
8
4
5
5
Equations of horizontal and vertical lines
The equation of a horizontal line with y-intercept (0,c) is
________________
__________
The equation of a vertical line with x-intercept (c,0) is
Write the equation for each line displayed. The scale on both axes is 1.
Identifying the slope and y-intercept in linear equations
Identify the slope and y-intercept in each equation. It may be necessary to transform the
equation into slope intercept form.
f(x) = x – 4
-3x + 2y = 5
y = -3
x=4
Parallel Lines: Two ________________________lines are parallel if the have the same
__________________ but different ______________________________________.
Chapter 3
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Perpendicular Lines: Two non-vertical lines with non-zero slopes m1 and m2 are
perpendicular if the ____________________of their slopes is _______
Pairs of perpendicular slopes
m1 -2
3/4 -1
0.25
m2 1/2
On the same set of axes, graph
y1   2 x  3
y2 
1
x 1
2
State whether each pair of equations represent parallel lines, perpendicular lines or
neither. (Remember, examine the slopes. Their product must be -1)
2 x  3 y 1
x 4y  3
4x  6 y   3
3 x  12 y  5
Slope in applications - Slope can be interpreted as a rate of change of a quantity.
In 1990 about 1.1 million SUV's were sold. In 1991 about 1.4 million were sold. Let x =
0 represent 1990. If this trend is linear, write a model to represent it and predict the
number of SUV's sold in 2000.
Chapter 3
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Section 3.5 - Equations and Linear Models, Parallel and Perpendicular Lines
Review
Write an equation in slope-intercept form for the line with slope -0.5 and vertical
intercept of 1.75. Does (2, 0.75) lie on this line?
Find the equations of the vertical line and the horizontal line passing through (2,-3).
Vertical line _____________
Horizontal line _____________
Give the parallel and perpendicular slopes to y = -3x + 5,
Parallel slope _______
Perpendicular slope ________
Point-slope form of a linear equation
A line with slope m passing through (x1, y1) is given by
_____________
or
_____________
Write an equation in point-slope form for a line with a slope of -3 which goes through the
point (2,4). Transform the equation into slope-intercept form.
Write an equation in point-slope form of a line passing through (-4, -2) with a slope of ½.
Give the slope and a point on the line:
a) y - 1 = 3(x – 5)
b) y = -3(x – 5) + 7
c) y = 10
d) x = -5
f) g(x) = 3x
Chapter 3
e) f(x) = 5x + 3
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Two methods for finding the equation of a line through any two points
. (This skill of writing equations for lines through points becomes really important when we solve
linear application problems and can describe the information given as an ordered pair.)
Method 1: Using point-slope formula
1) Compute slope m
2) Substitute into point-slope form
3) Transform the equation to slope-intercept form
4) Check that both points lie on the line
Use point-slope form to find the
slope-intercept form of the equation
of the line passing through
(-1, 5) and (4, -2).
Use the point-slope form to write the
slope-intercept form of the equation
for the line through the points
(-3, 2) and (1,4).
Method 2: Using slope-intercept formula
1) Calculate slope m
2) Algebraically find the y-coordinate (b) of the y-intercept
3) Using m and b write an equation for the line in slope-intercept form.
4) Check that both points lie on the line
Use slope-intercept to find an equation for the line which goes through the points (-3, 5)
and (2, 4). Write your final equation in slope-intercept form.
Finding the equations of a line through a given point that is parallel to a given line
Steps: 1) Determine the parallel slope
2) Use steps 2 through 4 from method 1 or 2 to find the slope-intercept form
Find the slope-intercept form of a
line parallel to y = 2x + 5 which
Find the slope-intercept form of a
line parallel to y   1 x  2 which
3
passes through the point (-2, 3).
Chapter 3
passes through the point (-1,4).
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Mth 95
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Finding the equations of a line through a given point that is perpendicular to a given
line
Steps: 1) Determine perpendicular slope
2) Use steps 2 through 4 from method 1 or 2 to find the slope-intercept form
Find the slope-intercept form of a
line perpendicular to y = -3x + 1
which passes through the point (-1, 5).
Find the slope-intercept form of a
line perpendicular to y = 2/3x + 1
which passes through the point (4, 1).
Application: The projected annual cost of the average private college or university is
shown in the table. The cost includes, tuition, fees, room and board.
Year
2003
2008
Cost
$25,000
$40,000
a) Find the slope intercept form of a line that goes through these data points.
b) Find another equation for this data by letting x = 0 represent the year 2000.
c) Compare the equations
d) Interpret the slope as a rate of change
e) Using you equation from b, estimate the cost of private college in 2005.
Chapter 3
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