Transcript Algebra 2 - Mrs. Schneider MathClawson High School

Algebra 2
Chapter 2: Linear Relations and Functions
Section 2.1
Relations and Functions
Objectives

Analyze and graph relations.

Find functional values.
Vocabulary

Ordered Pair: A pair of
coordinates, written in the form
(x, y), used to locate any point on
a coordinate plane.

Cartesian Coordinate Plane:
composed of the x-axis (horizontal)
and y-axis (vertical), which meet
at the origin (0, 0) and divide the
plane into four quandrants.
Relation; Domain; Range

Relation: is a set of ordered pairs.

Relation:
{ (12, 28), (15, 30), (8, 20), (12, 20), (20, 50)}


Domain (of a relation): the set of
all first coordinates (x-coordinates)
from the ordered pairs.
Range (of a relation): the set of
all second coordinates (ycoordinates) from the order pairs.

Domain:
{8, 12, 15, 20}

Range:
{20, 28, 30, 50}
Function



A function is a special type of
relation. Each element of the
domain is paired with exactly one
element of the range.
A mapping shows how the members
are paired. An example is shown
to the right.
The example to the right is a
function; each element of the
domain is paired with exactly one
element of the domain. This is
called a one-to-one function.
Functions can be represented as 𝑓 𝑥 or 𝑔 𝑥 .
When speaking, we say “F of x” or “G of x”.

Relation:
{(12, 28), (15, 30), (8, 20)}
Domain
Range
12
28
15
30
8
20
Function or not?
Domain
Range
Range
Domain
-3
1
-1
0
2
1
3
2
4
4
5
Function
Function
Domain
Range
-3
0
1
1
5
6
NOT a Function
Relations: Discrete or Continuous?
Discrete
Discrete graphs contain a set of
points not connected.
Continuous
Continuous graphs contain a smooth line
or curve.
Note: You can draw the graph of a continuous relation
Without lifting you pencil from the paper.
Vertical Line Test

If no vertical line intersects a
graph in more than one point, the
graph represents a function.

If some vertical line intersects a
graph in two or more points, the
graph DOES NOT represent a
function.
Graphing Relations

See examples on pages 60 and 61 in your textbook.

When graphing, create a table of values.
Evaluate a function

Given 𝑓 𝑥 = 𝑥 2 + 2, find each value.
a.
f(-3)
𝑓 𝑥 =
𝑥2
+2
𝑓 −3 = (−3)2 +2
𝑓 −3 = 9 + 2
𝑓 −3 =11
b. f(3z)
𝑓 𝑥 = 𝑥2 + 2
𝑓 3𝑧 = (3𝑧)2 +2
𝑓 3𝑧 = 9𝑧 2 + 2
HOMEWORK…..A#2.1

Assigned on Friday, 9/20/13

Due on Monday, 9/23/13

Pages 62-63 [#13-20 all, 24, 34, 36, 40]
Section 2.2
Linear Equations
Section Objectives

Identify linear equations and functions.

Write linear equations in standard form and graph them.
Identify Linear Equations and Functions
A linear equation has no operations other than addition, subtraction, and multiplication
of a variable by a constant. The variables may not be multiplied together or appear in a denominator.
It does not contain variables with exponents other than 1. The graph of a linear equation is
always a line.
Linear Equations
NOT Linear Equations

5𝑥 − 3𝑦 = 7

7𝑎 + 4𝑏2 = −8

𝑥=9

𝑦 = 𝑥+5

6𝑠 = −3𝑡 − 15

𝑥 + 𝑥𝑦 = 1

𝑦=

𝑦=
1
𝑥
2
1
𝑥
Identify Linear Equations

State whether each function is a linear function. Explain.
a.
𝑓 𝑥 = 10 − 5𝑥
b.
𝑔 𝑥 = 𝑥4 − 5
c.
ℎ 𝑥, 𝑦 = 2𝑥𝑦
5
d.
𝑓 𝑥 = 𝑥+6
e.
𝑔 𝑥 = −2𝑥 + 3
3
1
Standard Form

The standard form of a linear equation is…
𝐴𝑥 + 𝐵𝑦 = 𝐶
where A, B, and C are integers whose greatest common factor is 1, 𝐴 ≥ 0, and A and B
are not both zero.
Write each equation in standard form.
Identify A, B, and C.
a.
𝑦 = −2𝑥 + 3
b.
− 𝑥 = 3𝑦 − 2
3
5
c.
2𝑦 = 4𝑥 + 5
d.
3𝑥 − 6𝑦 − 9 = 0
Graphing with Intercepts

X-Intercept: the x-coordinate of the point at which it crosses the x-axis.
y=0

Y-Intercept: the y-coordinate of the point at which it crosses the y-axis.
x=0
Find the x-intercept and y-intercept of the
graph of 3𝑥 − 4𝑦 + 12 = 0. Then graph the
equation.
Find the x-intercept and y-intercept of the
graph of 2𝑥 + 5𝑦 − 10 = 0. Then graph the
equation.
HOMEWORK…..A#2.2

Assigned on Monday, 9/23/13

Due on Tuesday, 9/24/13

Page 107 [#16-22 all]
Section 2.3
Slope
Objectives for Section 2.3

Find and use the slope of a line.

Graph parallel and perpendicular lines.
Vocabulary

A rate of change measures how much a quantity changes, on average,
relative to the change in another quantity, often time.

The slope (m) of a line is the ratio of the change in y-coordinates to the
corresponding change in x-coordinates.
The slope m of the line passing through (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) is given by
𝑦 −𝑦
𝑚 = 𝑥2−𝑥1, where 𝑥1 ≠ 𝑥2
2
1
Find the slope of the line that passes through
(-1, 4) and (1, -2). Then graph the line.
Find the slope of the line that passes through
(1, -3) and (3, 5). Then graph the line.
Slope – tells the direction in which it
rises or falls.
Negative Slope
Zero slope
Family of graphs

A family of graphs is a group of graphs that displays one or more similar
characteristics.

The parent graph is the simplest of the graphs in a family.
Parent: y = x
Family: y = 3x + 2
y=x+2
Parallel Lines

In a plane, nonvertical lines with the same slope are parallel. All vertical
lines are parallel.
Graph the line through (-1, 3) that is parallel
to the line with equation 𝑥 + 4𝑦 = −4.
Graph the line through (-2, 4) that is parallel
to the line with equation 𝑥 − 3𝑦 =3.
Perpendicular Lines


Two lines are perpendicular if the
product of their slopes = −1.
When you have two perpendicular lines, their slopes are opposite reciprocals
of each other.
Slope of line AB:
C(-3,2)
A(2,1)
Slope of line CD:
D(1,-4)
B(-4,-3)
Graph the line through (-3, 1) that is perpendicular
to the line with equation 2𝑥 + 5𝑦 = 10.
Graph the line through (-6, 2) that is perpendicular
to the line with equation 3𝑥 − 2𝑦 = 6.
HOMEWORK…..A#2.3

Assigned on

Due on

Page 108 [#23-29 all]
Section 2.4
Writing Linear Equations
Objectives
After this section, you will be able to…

Write an equation of a line given the slope and a point on the line.

Write an equation of a line parallel or perpendicular to a given line.
Slope-Intercept Form of a Linear
Equation
𝑦 = 𝑚𝑥 + 𝑏
slope
y-intercept
Write an Equation Given Slope and a
Point

Write an equation in slope-intercept form for the lines that has a slope of
4
and passes through the point (3, 2).
3
Practice

Write and equation in slope-intercept form for the line that has a slope of
− 4 and passes through (−2, −2).
Graph an Equation in Slope-Intercept
Form

Graph the following equations:
𝑦=
4
𝑥
3
+2
𝑦 = −3𝑥 − 4
Point-Slope Form of a Linear Equation
Slope
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Given point
Write an Equation Given Two Points

What is the equation of the line through 2, 3 and −4, −5 ?
Procedure:
1. Find the slope.
2. Write an equation using slope
and one of the given points.
Write an Equation of a Perpendicular
Line

Write an equation for the line that passes through (3, 7) and is perpendicular
3
to the line whose equation is 𝑦 = 4 𝑥 − 5.
HOMEWORK…..A#2.4

Assigned on Thursday 9/26/13

Due on Friday 9/27/13

Page 108 [#30-34 all]
Section 2.5
Statistics: Using Scatter Plots
Objectives
After this section, you will be able to…

Draw scatter plots.

Find and use prediction equations.
Vocabulary

Bivariate Data:

Scatter Plot:
Speed (mph)
Calories
5
508
6
636
7
731
8
858
Scatter Plot Correlations
Prediction Equations

Line of Fit:

Prediction Equation:

To find a line of fit and prediction equation:
Find and Use a Prediction Equation
HOUSING: The table below shows the median selling price of new, privatelyowned, one-family houses for some recent years.
Year
1994
1996
1998
2000
2002
2004
Price
($1000)
130.0
140.0
152.5
169.0
187.6
219.6
Draw a Scatter Plot and a line of fit for the
data. How well does the line fit the data?
250
Price ($1000)
230
210
190
170
150
130
110
0
2
4
6
8
10
Years since 1994
Year
1994
1996
1998
2000
2002
2004
Price
($1000)
130.0
140.0
152.5
169.0
187.6
219.6
Find a prediction equation. What do the
slope and y-intercept indicate?
Predict the median price in 2014.
How accurate does the prediction appear to be?
PRACTICE

The table shows the mean selling price of new, privately owned one-family
homes for some recent years. Draw a scatter plot and line of fit for the data.
Then find a prediction equation and predict the mean price in 2014.
Year
1994
1996
1998
2000
2002
2004
Price
($1000)
154.5
166.4
181.9
207.0
228.7
273.5
1994
1996
1998
2000
2002
2004
Price
($1000)
154.5
166.4
181.9
207.0
228.7
273.5
Price ($1000)
Year
Years since 1994
Practice workspace
HOMEWORK…..A#2.5

Assigned on Monday 9/30/13

Due on Tuesday 10/1/13

Page 89 [#3-9 all]