Download VOCABULARY Relation: Domain: Range: Function:

2.1 Represent Relations and Functions
Standards:
Anchors:
2.8.8I, 2.8.11RST
M11.D.1.1.1, M11.D.1.1.2, M11.D.1.1.3, M11.D.2.1.2, M11.D.3.1.2
OBJECTIVE: TLW write relations as sets, mappings, tables, graphs and equations
TLW identify and evaluate functions
VOCABULARY
Relation:
Domain:
Range:
Function:
REPRESENTING RELATIONS
A relation can be represented in the following ways:
Ordered Pairs
(2, 2)
(2, 2)
(0, 1)
(3, 1)
Table
x
y
2
2
2
2
0
1
3
1
Graph
Mapping Diagram
Input
2
0
3
Output
2
2
1
Function: is a relation for which each input has exactly
one output.
If any input of a relation has more than one output, the
relation is NOT a function.
Identify functions
1) Tell whether each relation is a function. Explain.
a. Input Output
b. Input Output
2
1
3
6
1
0
4
2
1
2
2
0
2
3
Is the relation given by the ordered pairs
(5, 2), (3, 1), (0, 0), (0, 2) and (0, 5) a function?
Explain.
VERTICAL LINE TEST
Function
Not a function
Is the relation represented by the graph a function?
Explain.
a.
b.
Classify and evaluate functions
Linear function in x-y notation y = mx + b
Linear function in function notation f(x) = mx + b
Linear function has the variable or variables:x and y to the
first power.
Tell whether the function is linear. Then evaluate the
function when x = 3.
a. f(x) = 6x + 10 b. g(x) = 2x2 + 4x 1
Use the vertical line test to tell whether the relation is a
function.
Tell whether the function is linear. Then evaluate the
function when x = 1.
1.
f(x) = 2x3 + 6  x
2.
g(x) = 4x + 9
Homework: Skill 2.1
2.2 Slope and Graphing Linear Equations
Standards:
Anchors:
2.8.8DFGHI, 2.8.11EKLNQS, 2.11.8B
M11.C.3.1.2, M11.D.2.1.2, M11.D.3.1.1, M11.D.3.2.1, M11.D.3.2.3
OBJECTIVE: TLW calculate slope
TLW identify lines as intersecting, parallel, or perpendicular
TLW graph lines using the slope-intercept method
TLW graph lines using the x-y intercept method
TLW graph horizontal and vertical lines.
VOCABULARY
Slope: of a nonvertical line is the ratio of vertical change (the
rise) to horizontal change (the run)
Parallel: two lines in a plane never intersect
Perpendicular: two lines in a plane intersect to form a right
angle
Rate of change: or slope is how much one quantity changes,
on average relative to the change in another quantity
SLOPE OF A LINE
Words
The slope m of a nonvertical line is the ratio of __________
change (the rise) to _______ change (the run).
Algebra
m
Graph
y2  y1

x2  x1
Example 1
Find slope
What is the slope of the line passing through the points (1,
3) and (6, 7)?
Let (x1, y1)  (1, 3) and (x2, y2)  (6, 7).
CLASSIFICATION OF LINES BY SLOPE
Positive
slope
Rises
from
left to
right
Negative Zero
Undefined
slope
slope
slope
Falls
Horizontal Vertical
from left
to right
Classify lines using slope
Without graphing, tell whether the line through the given
points rises, falls, is horizontal, or is vertical.
a.
(6, 2), (1, 3)
b.
(2, 1), (2, 2)
Complete the following exercises.
1. Find the slope of the line passing through the points (4, 2)
and (7, 9).
2.
Without graphing tell whether the line through the points
(3, 2) and (1, 4) rises, falls, is horizontal, or is vertical.
SLOPES OF PARALLEL AND PERPENDICULAR
LINES
Consider two different nonvertical lines l1 and l2 with slopes
m1 and m2.
Parallel lines The lines
are parallel if and only if
they have the______
Perpendicular lines The
lines are perpendicular if
and only if their slopes
are________________
Classify parallel and perpendicular lines
Tell whether the lines are parallel or perpendicular.
Line 1: through (3, 1) and (2, 5)
Line 2: through (3, 4) and (3, 1)
Tell whether the lines are parallel, perpendicular, or
neither.
Line 1: through (1, 0) and (3, 4)
Line 2: through (24, 6) and (22, 5)
VOCABULARY
y-intercept
Slope-intercept form
Standard form of a linear equation
x-intercept
USING SLOPE-INTERCEPT FORM TO GRAPH AN
EQUATION
Step 1 Write the equation in ______________ form by
solving for y.
Step 2 _______ the y-intercept b and use it to plot the point
(0, b) where the line crosses the y -axis.
Step 3 Identify the ________ m and use it to plot a second
point on the line.
Step 4 ________ a line through the two points.
Graph an equation in slope-intercept form
Graph y = 32 x + 1.
USING STANDARD FORM TO GRAPH AN
EQUATION
Step 1 Write the equation in standard form.
Step 2 Identify the x-intercept by letting ___ = 0 and solving
for ____. Use the x-intercept to plot the point where the
line crosses the______.
Step 3 Identify the y-intercept by letting ____ = 0 and
solving for ____. Use the y-intercept to plot the point
where the line crosses the _______.
Step 4 Draw a line through the two points.
Graph an equation in standard form
Graph 2x + 3y = 12.
HORIZONTAL AND VERTICAL LINES
Horizontal lines The graph of y = c is the horizontal line
through (____,___).
Vertical lines The graph of x = c is the vertical line through
(____,____).
Graph horizontal and vertical lines
a. Graph y = 1
b. Graph x = 2.
Graph the equation.
1. y = 2x + 2
3. 4x + 2y = 8
4
2. y = 3 x 4
4. 5x + 3y = 15
5. y = 4
Homework: Skill 2.2
6. x = 2
2.3 Write Equations of Lines
Standard:
Anchor:
2.8.11L
M11.D.3.2.2
OBJECTIVE: TLW write linear equations when given:

slope and a point

two points

parallel to a given line and a point

perpendicular to a given line and a point
Given slope m and y-intercept
Use slope-intercept form:
y = mx + b
Given slope m and a point(x,y)
Use point-slope form:
y – y1 = m (x – x1)
Given points (x1,y1) and (x2, y2)
Find slope then
use point-slope form
WRITING AN EQUATION OF A LINE
1. find the slope of the line
2. use the slope and the point to set up equation
using the point slope form
3. change formula to standard form
Write an equation given the slope and y-intercept
Write an equation of the line shown.
Write an equation given the slope and a point
Write an equation of the line that passes through (2, 1)
and has a slope of 2.
Write equations of parallel or perpendicular lines
Write an equation of the line that passes through (1, 1)
and is (a) parallel to, and (b) perpendicular to, the line y =
2x + 3.
Write an equation given two points
Write an equation of the line through (3, 1) and (2, 3).
Write an equation of the line.
1.
2.
3.
Through (1, 5) with a slope of 2
Through (2, 3) and (a) parallel and (b) perpendicular to
y = 4x  6
4.
Through (6, 2) and (3, 2)
Homework: Skill 2.3
2.4 Draw Scatter Plots and Best Fitting Lines
Standards: 2.6.8AC, 2.6.11ACD, 2.8.11A
Anchors: M11.D.1.1.1, M11.E.1.1.1, M11.E.4.2.2
OBJECTIVE: TLW identify correlations
TLW find a line of best fit with and without a calculator
VOCABULARY
Scatter plot- is a graph of a set of data pairs (x,y)
Positive correlation – the y value tends to increase as x
increases
Negative correlation – the y value tends to decrease as x
increases
Correlation coefficient – denoted r is a number from -1 to 1
that measures how well a line fits a set of data points.
r is near 1 points lie near line with positive slope
r is near -1 points lie near line with negative slope
r is near 0 points do not lie near any line.
Best-fitting line – is the line that lies as close as possible to
all the data points
if correlation coefficient for a set of data is near ±1 the data
can be reasonably modeled by a line.
Estimate correlation coefficients
For each scatter plot, describe the correlation shown and
tell whether the correlation coefficient is closest to 1,
0.5, 0, 0.5, or 1.
a.
b.
Solution
a. The scatter plot shows a ____________ correlation. So, the
best estimate
given is r = ____.
b.
The scatter plot shows a ___________ correlation. So, r is
between ___ and ___ but not too close to either one. The
best estimate given is r = ____.
APPROXIMATING A BEST-FITTING LINE
Step 1 Draw a _________ of the data.
Step 2 Sketch the _____ that appears to follow most closely
the trend given by the data points. There should be about
as many points ______ the line as _______ it.
Step 3 Choose _________ on the line, and estimate the
coordinates of each point.
Step 4 Write an _________ of the line that passes through
the two points from Step 3.
Approximating a best-fitting line
The table below gives the number of people y who
attended each of the first seven football games x of the
season. Approximate the best-fitting line for the data.
x
1
2
3
4
y 722 763 772 826
1. Draw a ___________.
5
6
7
815
857
897
Be sure that about the
same number of
points lie above your
line of fit as below it
2.
Sketch the best-fit line.
3.
Choose two points on the line. For the scatter plot shown,
you might
choose(1, _____ ) and (2, _____ ).
4.
Write an equation of the line.
Use a line of fit to make predictions
Use the equation of the line of best fit from Example 2 to
predict the number of people that will attend the tenth
football game.
For each scatter plot (a) tell whether the data has
positive correlation, negative correlation, or no
correlation, and (b) tell whether the correlation
coefficient is closest to 1, 0.5, 0, 0.5, or 1.
1.
3.
The table gives the average class score y on each chapter
test for the first six chapters x of the textbook.
x
1
2
3
4
5
6
y 84 83 86 88 87 90
a. Approximate the best-fitting line for the data.
b. Use your equation from part (a) to predict the test score
for the 9th test that the class will take.
Homework: Skill 2.4
2.5 Use Absolute Value Functions
Standards: 2.5.11ABCD, 2.8.11RST
Anchors: M11.A.2.2.1
Objectives:
TLW graph absolute value functions
TLW graph using transformations, translations, and reflections
VOCABULARY
Absolute value function
Vertex of an absolute value graph
Transformation
Translation
Reflection
The highest or lowest point on the graph of an absolute value
function is called the _____________________________.
The vertex of the graph y  | x | is (___, ___).
TRANSFORMATIONS OF GENERAL GRAPHS
Graph functions of the form y = a | x |
1
Graph (a) y = 3 x and (b) y = 2 | x |. Compare each
graph with the graph of y  | x |.
Graph the function. Compare the graph with y = | x |.
1. y = 3 | x |
Graph a function of the form y  a | x  h |  k
Graph y = 3 | x  2 |  1. Compare the graph with the
graph of y = | x |.
Write an absolute value equation
Write an equation of the graph shown.
2.
Graph the function y  
graph with the graph of
y = | x |.
1
2
| x  1 |  2. Compare the
Write an equation of the graph shown.
Homework: Skill 2.5
2.6 Other Functions
Standards: 2.5.11ABCD, 2.8.11RST
Anchors: M11.D.1.1.1, M11.D.1.1.2, M11.D.1.1.3
OBJECTIVE: TLW evaluate, graph and write piecewise (compound) functions
TLW evaluate, graph and write step functions
Vocabulary:
 Compound function
 Step function
Compound Functions
Use the function below to evaluate and graph:
3x  2
f ( x)  
x  2
Find f(-3)
x0
x0
f(2) f(0)
Graph the function
Write a function for the following problem:
A silk screen shop has the following price schedule for
silk-screen t-shirts.
 An initial charge of $25 to create the silk-screen
 $10.50 per shirt for orders of 25 or fewer shirts
 $9.75 per shirt for orders of more than 25 shirts
Step Functions
Use the function below to evaluate and graph:
10
4

f ( x)  
1
3
x 1
1 x  2
2 x4
x4
Find f(-3)
f(2) f(10)
Graph the function
Homework: Skill 2.6
2.7 Graph Linear Inequalities in Two Variables
Standard:
Anchor:
2.8.8EFG, 2.8.11KN
M11.D.2.1.2
Objective:
TLW graph linear inequalities in two variables with and without a graphing calculator
Check whether the ordered pairs (a) (3, 2) and (b) (1, 4)
are solutions of
4x + 2y > 6.
Check whether the ordered pair is a solution of 2x  y  8.
1.
2.
(6, 2)
(3, 1)
GRAPHING A LINEAR INEQUALITY
To graph a linear inequality in two variables, follow these
steps:
Step 1 Graph the boundary line for the inequality.
Step 2 Test a point ________ the boundary line to determine
whether it is a solution of the inequality.
Graph a linear inequality with one variable
Graph y < 1 in a coordinate plane.
Graph a linear inequality with two variables
Graph 3x  2y < 6 in a coordinate plane.
Graph an absolute value inequality
Graph y > 3 x  1+ 2 in a coordinate plane.
Graph the inequality in a coordinate plane.
3. x < 2
5. 9x + 3y > 9
4.y  x + 2
6. y  2 |x + 2|  l
GRAPHING INEQUALITIES ON TI-84
Use Apps – Inequalities
Homework: Skill 2.7