Chapter 2: Linear Equations
and Functions
2.1 Functions and Their
Graphs
Vocabulary
• Relation – pairing of input vs. output values
• Domain – input values
• Range – output values
• Function – exactly one output for each input
• Identify the domain and range. Then determine
if the relation is a function.
Input
Output
-3
3
1
-2
1
1
4
4
Identify the domain and range. Then determine if
the relation is a function.
Input
Output
-3
3
1
1
3
1
4
-2
Vocabulary
• Ordered pairs –(x,y) coordinates
• Coordinate plane – divided into four
quadrants by x and y axis that intersect at
the origin.
Graph the following points.
•
•
•
•
(-3,3)
(1,-2)
(1,1)
(4,4)
State the Coordinates
Vertical Line Test for Functions
• A relation is a function if and only if no
vertical line intersects the graph of the
relation at more than one point.
Vocabulary
• Equation – represents a function
• Solution – ordered pair which makes the
equation true
• Independent variable – input variable
• Dependent variable – output variable
• Graph – coordinates are solutions of the
equation
Graphing Equations in Two Variables
• 1. Make a table of values
• 2. Graph enough solutions to recognize a
pattern
• 3. Connect the points with a line or curve.
Vocabulary
• Linear function –
y  mx  b
• Function notation - f ( x )  mx  b
Graph the function
y  2x  1
2.2 Slope and Rate of Change
Vocabulary
• Slope – ratio of vertical change to
horizontal change (rise over run)
• Parallel – lines that never intersect
• Perpendicular – lines that form a right
angle when they intersect
Slope of a Line
• Slope of a nonvertical line passing through two
points ( x , y ) and ( x , y )
1
1
m
2
y2  y1
x2  x1

2
rise
run
• Find the slope of the line passing through
the two points:
(-3,5) and (2,1)
Classification by Slope
•
•
•
•
Positive slope rises
m>0
Negative slope falls
m<0
Zero slope is a horizontal line
m=0
Undefined slope is a vertical line
Classify the Line Using Slope
• A. (3,-4) and (1,-6)
• B. (2,-1) and (2,5)
Slopes of Parallel and
Perpendicular Lines
• Parallel lines
m1  m2
• Perpendicular lines
m1  
1
m2
or m1m2  1
• Tell whether the lines are parallel, perpendicular
or neither.
• A. Line 1: (-3,3) and (3,-1)
Line 2: (-2,-3) and (2,3)
B. Line 1: (-3,1) and (3,4)
Line 2: (-4,-3) and (4,1)
2.3 Quick Graphs of Linear
Equations
Vocabulary
• y-intercept – intersection of y-axis (0,b)
• Slope-intercept form – y = mx+b, m is
slope and b is y-intercept
• Standard form – Ax+By = C
• x-intercept – intersection of x-axis
Graphing Equations in
Slope-Intercept Form
•
•
•
•
1. Solve for y.
2. Find y-intercept and plot on graph.
3. Use slope to plot a second point on line.
4. Draw a line through the two points.
Graph
y  x2
3
4
The number of gallons (g) of water in your
storage tank is given by g = 500-20t, with t in
days. Graph this relation. What is the daily rate
of water usage? How many days will pass
before the tank is empty?
Graphing Equations in Standard Form
1. Write the equation in standard form.
2. Find the x-intercept by letting y = 0 and
solving for x. Plot the x-intercept.
3. Find the y-intercept by letting x = 0 and
solving for y. Plot the y-intercept.
4. Draw a line through the two points.
Graph 2x+3y = 12
Horizontal and Vertical Lines
• Horizontal line – graph of y = c through (0,c)
• Vertical line – graph of x = c through (c,0)
Graph
• A. y = 3
• B. x = -2
The school band is selling sweatshirts and Tshirts to raise money. The goal is to raise
$1200. Sweatshirts sell for a profit of $2.50
each and T-shirts for $1.50 each. Explain how
many sweatshirts or T-shirts the band can sell
to reach their goal.
2.4 Writing Equations of Lines
Vocabulary
• Direct variation – y = kx with k as a
constant, y varies directly with x. Graph is
a line through the origin.
• Constant of variation – constant k, k≠0
Writing an Equation of a Line
• Slope-intercept form – given the slope m and yintercept b, use this equation:
y = mx + b
Writing an Equation of a Line
• Point-slope form – given the slope m and a
point ( x1 , y1 ), use this equation:
y  y1  m( x  x1 )
Writing an Equation of a Line
• Given two points ( x1 , y1 ) and ( x2 , y2 ) , use the
formula
y2  y1
m
x2  x1
to find the slope m. Then use the point-slope
form with this slope and either of the given
points to write an equation of the line.
Checklist
1. Identify what information is given.
2. Match the given information with the
three equations.
3. Choose which equation is needed to
write the equation.
Write an equation for the line.
Write an equation for the line.
1. Line that passes through the (2,3) and
has a slope of -2.
2. Line passing through (-3,-4) and (1,-6)
Write an equation for the line.
• Line that passes through (3,2) and is parallel to
the line
y  3x  2
• Line that passes through (3,2) and is
perpendicular to the line y  3x  2
• The variables x and y vary directly, and
y = 12 when x = 3. Write and graph an
equation relating x and y.
Using Direct Variation
• The variables x and y vary directly, and
y = 12 when x = 3. Find y when x = -2.
In 1990 retail sales at bookstores were about
$7.4 billion. In 1997 retail sales at bookstores
were about $11.8 billion. Write a linear model
for retail sales s (in billions) at bookstores from
1990 through 1997. Let t represent the years
from 1990. Then estimate the retail sales in
2012.
In 1990 retail sales at bookstores were about $7.4
billion. In 1997 retail sales at bookstores were about
$11.8 billion. Write a linear model for retail sales s (in
billions) at bookstores from 1990 through 1997. Let t
represent the years from 1990. Then estimate the
retail sales in 2012. m  11.87.4  0.629
1997 1990
s  7.4  0.629(t  1990)
s  0.629(22)  7.4
s  0629
. t  7.4
s  $21.2 billion
2.6 Linear Inequalities in
Two Variables
Vocabulary
• Linear inequality -
Ax  By  C
Ax  By  C
Ax  By  C
Ax  By  C
• Solution – ordered pair (x,y) that makes the
inequality true
• Graph – graph of all solutions of the inequality
• Half-planes – boundary line of inequality that
divides the coordinate plane, one half is shaded
and the other is blank
Graphing a Linear Inequality
• The graph of a linear inequality in two variables
is a half-plane. To graph, follow these steps.
• 1. Graph the boundary line of the inequality.
Use a dashed line for < or > and a solid line for
≤ or ≥.
• 2. Pick a point not on the line and test to see if it
is a solution. If it is a solution, shade the side
containing that point. If it’s not a solution, shade
the side not containing the point.
Graph y < -2
Graph x ≤ 1
Graph y < 2x
Graph 2x – 5y ≥ 10
You have $200 to spend on CD’s and music
videos. CD’s cost $10 each and music videos
cost $15. Write a linear inequality to represent
the number of CD’s and music videos you can
buy. Then graph the inequality.
2.7 Piecewise Functions
Vocabulary
• Piecewise functions – combination of functions
corresponding to different domains
• Step function – piecewise function resembling
stairsteps
– Greatest integer function -
g( x)  x
Evaluate f(x) when x = 0, 2, 4
x  2, if x  2
R
f ( x)  S
T2x  1, if x  2
Graph the piecewise function.
R
f ( x)  S
T x  3,
1
2
x  23 , if x  1
if x  1
Graph
x  , if x  2
R
f ( x)  S
T x  1, if x  2
2
3
2
3
Graph the stepwise function.
1, if 0  x  1
R
||2, if 1  x  2
f ( x)  S
||3, if 2  x  3
T4, if 3  x  4
Graph
1, if - 4  x  3
R
||2, if - 3  x  2
f ( x)  S
||3, if  2  x  1
T4, if  1  x  0
Write the equation for the graph
Write the equation for the graph
Shipping costs $6 on purchases up to $50, $8
on purchases over $50 up to $100, and $10 on
purchases over $100 up to $200. Write a
piecewise function for these charges. Give the
domain and range.
2.8 Absolute Value Functions
Vocabulary
• Vertex – corner point of the graph (base of
the V)
Vertex
Graphing Absolute Value Functions
•
y  a xh k
• Vertex (h,k) and line of symmetry at x = h
• Opens up if a > 0, and opens down if a < 0
• Graph is wider than y = lxl if lal < 1
• Graph is narrower than y = lxl if lal > 1
Graph
y  x 1  2
Graph y   x  2  3
Write an Equation of the Graph
Write an Equation of the Graph
• The front of a roof with its outer edges 8 feet above
ground can be modeled by the following equation with
x and y in feet. Graph the function, then identify the
2
domain and range.
y  x  9  14
3