Transcript Y - Bibb County Schools

Unit 4
Graphing and Analyzing Linear
Functions
Michelle A. O’Malley
League Academy of Communication Arts
Greenville, South Carolina
Standards for Learning Goal 4.1
EA1.5: Demonstrate an understanding of
algebraic relationships by using a variety of
representations (including verbal, graphic,
numerical, and symbolic)
 EA3.1: Classify a relationship as being
either a function or not a function when
given data as a table, set or ordered pairs,
or graph.

Essential Question for Learning
Goal 4.1
 What
does a function look like?
Learning Goal 4.1 Notes



In order to determine whether a relationship is
linear, you should focus on the rate of change in
the relationship.
Linear relationships are characterized by a
constant change in one variable associated with
a constant change in the other variable.
That is, for each unit change in the independent
quantity (variable), there is a constant change in
the dependent quantity (variable).
Learning Goal 4.1 Notes





A constant rate of change is what makes the line
“straight.”
A constant increase in one variable compared to
the other is associated with straight lines having a
positive slope.
A constant decrease in one variable as compared
to the other is associated with straight lines having a
negative slope.
Some linear functions are proportional and take
the form of y=mx.
Some linear functions are non-proportional and
take the form y=mx + b.
Learning Goal 4.1 Example 1: Linear
or Non Linear?

Phil is making a 3 foot by 4 foot banner for
the math club. Realizing that the banner is
too small, he decides to increase each
side. Phil must decide how the new
dimensions will affect the cost of the
materials. (cost versus area)
Non-Linear
because the change in area is not constant
Learning Goal 4.1 Example 2: Linear
or Non Linear?

A scuba diver is 120 feet below sea level.
She knows that to avoid suffering from the
bends, she must come up at a rate of 7 feet
per minute (depth versus time).
Linear
because the rate of change is constant
Learning Goal 4.1 Example 3: Linear
or Non Linear?



The pattern in the first table is linear: The
constant rate of change in the y is zero and
the function is y=3.
The second pattern is also linear; the
constant rate of change in y is -3 (for a unit
change in x) and the function is y=-3x+1.
Note: when finding the rate of change if
the x-values do not increase in equal
increments, as long as the rates of change
(change in y/change in x) are equivalent
(i.e. (-9/3 = -3/1 = -6/2 = -3), the function is
linear.
The third pattern is not linear: for each 1
unit increase in x, the change in y is not
constant.
X
Y
X
Y
X
Y
-2
3
-4
13
-2
4
-1
3
-1
4
-1
1
0
3
0
1
0
0
1
3
1
-2
1
1
2
3
3
-8
2
4
7
-20
3
9
3
3
Table 1
Table 3
Table 2
Learning Goal 4.1 Essential Knowledge



Students should be able to identify that a
constant rate of change is the criteria used to
determine whether or not a relationship is linear.
Students should be able to determine if a
relationship given in tabular form represents a
linear function.
Students should be able to determine if a
relationship given in verbal form represents a
linear function.
Standards for Learning Goal 4.2
EA-5.6 Carry out a procedure to determine
the slope of a line from data given tabularly,
graphically, symbolically, and verbally.
 EA-5.7 Apply the concept of slope as a rate
of change to solve problems.

Essential Question for Learning
Goal 4.2
 What
is slope?
Learning Goal 4.2 Notes




Slope is usually represented by the letter m (from
the French, monter, which means “to go up”).
A constant rate of change is called the slope of a
line.
Slope is the ratio of vertical change to horizontal
change between points on the graph.
Slope is defined as m = (y2 – y1)/(x2 – x1), where
(x1, y1) and (x2, y2) are points on the line.
Learning Goal 4.2 Notes

Slope should also be considered in terms
of a ratio.



Slope is the ratio for the differences in the x
and y coordinates of two points on the line.
The ratio of vertical rise to horizontal run
(m=rise/run)
Slope is a rate of change.
Learning Goal 4.2 Notes




Slopes can be positive, which means rising to
the right.
Slopes can be negative, which means falling to
the right.
Slopes can be zero, which means the line will
be horizontal.
Slopes can be Undefined, which are
sometimes referred to as “no slope” which
should not be confused with zero slope.
Learning Goal 4.2 Notes




For a line to have a positive slope, as the independent
quantity increases (or decreases), the dependent quantity
increases (or decreases).
For a line to have a negative slope, as the independent
quantity increases (or decreases), the dependent quantity
decreases (or increases).
For a line to have a zero slope (equation y=constant), there is
no change in the dependent quantity associated with any
change in the independent quantity. (horizontal line)
For a line to have an undefined slope (equation x=constant),
there is no change in the independent quantity associated with
changes in the dependent quantity (no division by zero)
(vertical line)
Learning Goal 4.2 Notes

Slope has a proportional
nature.

For example, for Y=3x+1


For each unit increase in
x, y increases by 3
If x decreases by 10, y
will decrease by 30
X
Y=3x + 1
1
4
2
7
3
10
4
13
5
16
6
19
7
22
8
25
9
28
10
31
Learning Goal 4.2 Essential Knowledge




Students should know how to find slope when given a
table, graph, set of ordered pairs or algebraic
representation.
Students should be able to determine whether the slope
is positive, negative, zero, or undefined (not a function)
when given the graph of a line.
Students should be able to sketch an appropriate graph
when given slope values that are positive, negative,
zero, or undefined.
Students should be able to interpret the slope as the
rate of change in the context of a problem.
Standards for Learning Goal 4.3



EA 1.3 Apply algebraic methods to solve
problems in real-world contexts.
EA 1.5 Demonstrate an understanding of
algebraic relationships by using a variety of
representations (including verbal, graphic,
numerical, and symbolic).
EA 3.6 Classify a variation as either direct
or inverse.
Essential Questions for Learning
Goal 4.3
 How
do I distinguish between
direct and inverse variation?
Learning Goal 4.3 Notes

Direct Variation



Two variables, x and y, vary directly if there is a
nonzero number k such that y = kx.
Just because one quantity increases when the other
increases does not mean that x and y vary directly.
When one quantity always changes by the same
factor as another, the two quantities are in direct
proportion; “k” is the constant of proportionality as
well as the slope of the linear function.
Learning Goal 4.3 Notes

Inverse Variation




Two variables, x and y, vary inversely if there is a non-zero
number k such that y = k/x, or xy = k (x ≠ 0)
Just because one quantity decreases as the other increases
does not mean that the two quantities are inversely
proportional.
When one quantity always decreases by the same factor as
the other increases, the two quantities are inversely
proportional.
If xy=9 and x is multiplied by 2, then y is divided by 2 in order
to preserve the constant 9.
Learning Goal 4.3 Notes




Although Direct Variation is a linear function, it will
be of the form y = mx, not y = mx + b because the yintercept must be zero.
Given that the y-intercept must be zero, the graph will
always go through the origin (0,0)
The Graphical representation of inverse variation is
a hyperbola (curved line).
Inverse variation – the product of the two values
remains constant as x increase, y decreases, or as x
decreases, y increases.
Learning Goal 4.3 Essential Knowledge





Students should be able to determine whether the data
demonstrates direct variation, inverse variation, or neither
when given a graph, table, or real world application.
Students should be able to find the constant of variation, k,
and write an equation that relates x to y when given function
values.
Students should be able to find the constant of variation
when given a direct or inverse variation graph.
Students should be able to sketch a graph of the function,
when given a direct or inverse variation equation.
Students should be able to write an inverse or direct
variation equation when given a verbal description.
Standards for Learning Goal 4.4



EA 1.5 Demonstrate an understanding of
algebraic relationships by using a variety of
representations (including verbal, graphic,
numerical, and symbolic).
EA 1.6 Understand how algebraic relationships
can be represented in concrete models, pictorial
models, and diagrams.
EA 4.6 Represent linear equations in multiple
forms (including point-slope, slope-intercept, and
standard).
Essential Questions for Learning
Goal 4.4
 How
do I translate between the
various algebraic forms of a
linear function?
Learning Goal 4.4 Notes

Equations of the form ax + by = c
(standard form) and/or y = mx + b
(Slope-intercept form), or any equations
that can be transformed into either of these
two forms are linear.
Learning Goal 4.4 Notes


In the equation ax + by = c, if a = 0 then the
equation becomes y = c/b; this is the
equation of a horizontal line.
In equation ax + by = c, if b = 0 then the
equation becomes x = c/a; this is the
equation of a vertical line.
Learning Goal 4.4 Notes


A solution of an equation in two variables, x
and y, is an ordered pair (x, y) that makes
the equation true.
The graph of an equation in x and y is the
set of all points (x, y) that are solutions of
the equation.
Learning Goal 4.4 Quiz
2x = y
Y=2
X 2 = 3y
X/2+y/3=4
Y – 4 = 2x / 3
7x – 3x = 10
7x – 3x = 10y
Y=2/x
Xy = 5
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
No
No
No
Learning Goal 4.4 Essential Knowledge


Students should be able to determine if an
equations represents a linear function.
Students should be able to translate
between slope-intercept {y=mx + b}, pointslope {y – y1 = m(x – x1)}, and standard
forms {Ax + By = C} of linear functions.
Standards for Learning Goal 4.5



EA 1.1 Communicate a knowledge of algebraic
relationships by using mathematical terminology
appropriately.
EA 1.3 Apply algebraic methods to solve
problems in real-world contexts.
EA 5.5 Carry out a procedure to determine the
x-intercept and y-intercept of lines from data
given tabularly, graphically, symbolically, and
verbally.
Essential Questions for Learning
Goal 4.5
 What
do intercepts mean?
Learning Goal 4.5 Notes

Essential knowledge and skills for this
unit:



Be able to find the x and y intercepts of al ine
given the equation, table, or graph
Be able to graph a line using the intercepts
Be able to interpret the real world meaning of
the x and y intercepts
Learning Goal 4.5 Notes

How do you find the x and y intercepts?

First you must have an equation

For example, y = -3x + 48
 The slope is -3
 The y intercept is 48 – you know this because you know in
y=mx + b, b is the y intercept. Also, if you replace x with a
0, your equation will change to y=-3(0) + 48, which gives
you y=48.
 The x intercept can be found by substituting 0 for y; for
example, 0 = -3x + 48, -48=-3x, divide by three to isolate x
and your answer would be x = 16, which is the x intercept.
Learning Goal 4.5 Notes


If you are looking at coordinate
pairs, (x,0) represents the x
intercept and (0,y) represents the
y intercept.
Which one of the numbers in the
table on the right represent the x
and y intercepts.
X
Y
0
2
2
7
4
0
8
-2
12 -7
Learning Goal 4.5 Notes



Intercepts cross the graph at a given point where
one value is zero and the other is either the y or x
value.
Vertical lines have no y intercepts and x is
constant. (Note: A vertical line is not a function
and the slope would be undefined)
Horizontal lines have no x intercept and y is
constant. (Note: A horizontal line is a function
and the slope would be zero.
Learning Goal 4.5 Notes

Today you have $48 left from money you
got for your birthday. On average you have
been spending $3 per day and you are not
planning on changing this spending pattern.

Generate an equation that represents your
spending patterns; clearly indicate what x and y
represent?
Learning Goal 4.5 Notes

Today you have $48 left from money you got
for your birthday. On average you have been
spending $3 per day and you are not planning
on changing this spending pattern.

Generate an equation that represents your spending
patterns; clearly indicate what x and y represent?

Answer:
 y will be the amount of birthday money $ and x will be the
number of days
 Y=-3x + 48
Learning Goal 4.5 Notes

Today you have $48 left from money you
got for your birthday. On average you have
been spending $3 per day and you are not
planning on changing this spending pattern.

Next, What is the slope of the equation? What
does the slope represent in general and in the
context of this problem?
Learning Goal 4.5 Notes

Today you have $48 left from money you got for
your birthday. On average you have been
spending $3 per day and you are not planning
on changing this spending pattern.

Next, What is the slope of the equation? What does
the slope represent in general and in the context of
this problem?

Answer:
 The slope is – 3, which is the amount of money spent per day
Learning Goal 4.5 Notes

Today you have $48 left from money you
got for your birthday. On average you have
been spending $3 per day and you are not
planning on changing this spending pattern.

Next, Identify the x intercept in this problem
situation and explain what it represents in this
situation.
Learning Goal 4.5 Notes

Today you have $48 left from money you got
for your birthday. On average you have been
spending $3 per day and you are not
planning on changing this spending pattern.

Next, Identify the x intercept in this problem
situation and explain what it represents in this
situation.

Answer:
 X intercept will be when the birthday money has been spent
– it is all gone.
 Y= -3x + 48, 0= -3x + 48, -48= -3x, isolate the variable by
dividing by -3, x = 16
Learning Goal 4.5 Notes

Today you have $48 left from money you
got for your birthday. On average you have
been spending $3 per day and you are not
planning on changing this spending pattern.

Next, Identify the y intercept in this problem
situation and explain what it represents in this
situation.
Learning Goal 4.5 Notes

Today you have $48 left from money you
got for your birthday. On average you have
been spending $3 per day and you are not
planning on changing this spending pattern.

Next, Identify the y intercept in this problem
situation and explain what it represents in this
situation.

Answer:
 The y intercept is the amount of birthday money that is
left at that particular day.
 Y = -3(0) + 48, y = 48
50
Graph of
45
Y = - 3x + 48
And Y intercept = 48
What does your
graph look like?
Birthday Money $ - y axis
X intercept = 16
40
35
30
25
20
15
10
5
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Number of Days – x axis
Purpose Statement based on
Algebra 1 Learning Goal 4.6

Seventh and Eighth grade Algebra 1 students will be
assessed on their ability to write and graph a linear
equation using the slope-intercept form, with and without
technology. Relate the slope-intercept form to
transformations of the parent function y=x so that the
teacher, parents, instructional coach, and principal can
determine if the students have met the required Algebra 1
state standards that are represented in the Graphing and
Analyzing Linear Equations Unit (Greenville County
School District, 2010).
Standards for Learning Goal 4.6




EA 1.7 Understand how to represent algebraic
relationships by using tools such as handheld
computing devices, spreadsheets, and computer
algebra systems (CASs).
EA 3.5 Carry out a procedure to graph parent functions
(including ).
EA 4.1 Carry out a procedure to write an equation of a
line with a given slope and a y-intercept.
EA 4.2 Carry out a procedure to write an equation of a
line with a given slope passing through a given point.
Standards for Learning Goal 4.6
(Continued)




EA 5.1 Carry out a procedure to graph a line when
given the equation of the line.
EA 5.2 Analyze the effects of changes in the slope, m,
and the y-intercept, b, on the graph of y = mx + b.
EA 5.3 Carry out a procedure to graph the line with a
given slope and a y-intercept.
EA 5.4 Carry out a procedure to graph the line with a
given slope passing through a given point.
Essential Questions for Learning
Goal 4.6
 How
do I use transformations to
graph linear equations in slope
intercept form?
Learning Goal 4.6 Notes


Slope-intercept form of the linear equation
should be approached as transformations
(slides, flips and stretches or compressions) to
a parent function y = x.
Characteristics of a parent function y = x




The x and y coordinates of ordered pairs are equal
and include but are not limited to (-1,-1), (0,0), (1,1),
(5/2,5/2), ETC.
Slope = 1/1 = 1
Passes through the origin
Bisect first and third quadrants creating 45 degree
angles
Learning Goal 4.6 Notes

Students should recognize changes in the
value of m (slope)


When m is positive, the result is a stretch or
compression of the parent function y = x.
When m is negative, think of this as two
transformations, a stretch or compression followed by
a reflection of the parent function about the x axis.

(Note: remember that slope can be plotted from any
point on a given line not just from the origin)
Learning Goal 4.6 Notes

Students should recognize changes in the
value of b (y intercept)



Changes in the value of b results in vertical
translations of the function y = mx.
If b is positive, the graph of y = mx shifts up b units
If b is negative, the graph of y = mx shifts down b
units
Learning Goal 4.6 Notes





Students are
to draw a
graph
representing
the following:
Y=X
Y = 2/3x
Y = -2/3x
Y = -2/3x + 4
Note: the above lines represent the changes in the parent function;
however, they are closely placed on the grid but not exactly.
Learning Goal 4.6 Notes

Practice problems:



1. If the graph of the equation y = 3x + 4 is shifted
down 9 units, what is the equation of the new graph?
Draw the graph of a linear equation with negative
slope that intersects the positive y axis and write the
equation. Explain why your graph satisfies the given
conditions.
A line passes through the point (4,2). Find at least
five possible equations for this line.
Purpose Statement based on
Algebra 1 Learning Goal 4.7

Seventh and Eighth grade Algebra 1 students
will be assessed on their ability to use two
points to graph and write the equation of the
linear function that contain the given points so
that the teacher, parents, instructional coach,
and principal can determine if the students have
met the required Algebra 1 state standards that
are represented in the Graphing and Analyzing
Linear Equations Unit (Greenville County
School District, 2010).
Standards for Learning Goal 4.7



EA 1.3 Apply algebraic methods to solve
problems in real-world contexts.
EA 4.3 Carry out a procedure to write an
equation of a line passing through two
given points.
EA 5.1 Carry out a procedure to graph a
line when given the equation of the line.
Essential Questions for Learning
Goal 4.7
 How
do I use real-world data
to write the equation of a line?
Learning Goal 4.7 Notes

Essential knowledge and skills for this
unit:


Students must be able to graph and write the
equation of a linear function given two points.
Students must when given the graph of a linear
function, choose two points and write the
equation of the function.
Can you find the
equation of the
line?
Since you can find the
slope of the line and
you have a y-intercept,
you can create an
equation using slope-
10
9
8
7
You can also use the
slope formula by picking
two points on the line.
6
5
intercept or point-slope.
4
Slope is -5/4 and the yintercept is 5 therefore,
the equation of the line
3
(0,5)
M=y2-y1/x2-x1
M=(5-0)/(0-4) = -5/4
2
is: y = -5/4x + 5
1
Rise = -5 and Run = 4
0
Therefore, the slope is -5/4
(4,0)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Learning Goal 4.7 Notes



Students need to make sure that they can find the equation of a
linear function by using slope-intercept as well as point-slope.
To write an equation in Slope-intercept (y=mx+b), you need the
slope and the y-intercept. When slope = -5/4 and the yintercept = 5 the slope intercept form is y= -5/4 + 5.
In point-slope you need two coordinate points



(0,5) and (4,0)
Step 1 find the slope using m=(y2-y1)/(x2-x1), m=(5-0)/(0-4) = -5/4
Next, choose one coordinate point and use the slope to create you
point-slope equation
 Y-y1=m(x-x1)
 Y-5=-5/4(x-0)
 Y-5=-5/4x
 Y=-5/4x + 5
Learning Goal 4.7 Notes

Sample Problem

A candle that has been burning for two minutes is
eight inches long. That same candle is seven
and one-quarter inches long after burning three
minutes. Assume that the candle will continue to
burn at this same rate. Define a variable for the
time since the candle was lit and use this variable
to write a function for the length of the candle.

The above highlighted information in yellow
represents the coordinate points for the linear
equation. (5,7.25) and (2, 8)
Learning Goal 4.7 Notes

Sample Problem (continued)

Now that you have two points, you can find the slope.


m=(y2-y1)/(x2-x1), m=(8-7.25)/(2-5) = -1/4
Next, use one coordinate point and the slope to write the
point-slope equation.

Y-y1=m(x-x1)
Y-8=-1/4(x-2)
Y-8=-1/4x + 1/2
Y-8 +8 = -1/4x + ½ +8
y = -1/4x + 8 ½

Therefore, the y-intercept is 8 ½ inches




Learning Goal 4.7 Notes

Sample Problem (continued)

A candle that has been burning for two minutes is eight inches long. That
same candle is seven and one-quarter inches long after burning three
minutes. Assume that the candle will continue to burn at this same rate.
Define a variable for the time since the candle was lit and use this variable to
write a function for the length of the candle.

Question 1: When will the candle completely burn itself out?
This is the x-intercept







Y=-1/4x + 8 ½
0=-1/4x +8 ½
-8 ½ = -1/4x
(-4/1)(-17/2) = (-1/4x)(-4/1)
68/2 = x
34 = x
The x-intercept is 34 minutes
Learning Goal 4.7 Notes

Sample Problem (continued)

A candle that has been burning for two minutes is eight inches long.
That same candle is seven and one-quarter inches long after burning
three minutes. Assume that the candle will continue to burn at this
same rate. Define a variable for the time since the candle was lit and
use this variable to write a function for the length of the candle.

Question 2: How long will the candle be 20
minutes after it was lit according to this algebraic
model?

Take you equation y=-1/4x + 8 ½ and put 20 in place
of x.
 Y=-1/4(20) + 8 ½
 Y=-5 + 8 ½
 Y= 3 ½ inches, which is the size of the candle after 20
minutes
Learning Goal 4.7 Notes

Sample Problem (continued)

A candle that has been burning for two minutes is eight inches long.
That same candle is seven and one-quarter inches long after burning
three minutes. Assume that the candle will continue to burn at this
same rate. Define a variable for the time since the candle was lit and
use this variable to write a function for the length of the candle.

Question 3: How long was the candle when it was
lit?

This would be your y-intercept which was 8 ½
inches
Learning Goal 4.7 Notes

Sample Problem (continued)

A candle that has been burning for two minutes is eight inches long.
That same candle is seven and one-quarter inches long after burning
three minutes. Assume that the candle will continue to burn at this
same rate. Define a variable for the time since the candle was lit and
use this variable to write a function for the length of the candle.

Question 4: When will the candle be half of its
original height?





17/4 – 17/2 = -1/4x + 17/2 – 17/2
17/4 – 34/4 = -1/4x
-17/4 = -1/4x
(-4/1)(-17/4) = (-1/4x) (-4/1)
17 = x
Can you graph the
equation
y=-1/4x + 8 1/2
What is the y
intercept?
9
(2,8)
8
7
Height of Candle
What is the
Slope of this
equation?
10
6
(5,7.25)
5
4
3
2
1
What is the x
intercept?
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34
minutes
Standards for Learning Goal 4.8






EA 4.2 Carry out a procedure to write an equation of a line with a
given slope passing through a given point.
EA 4.3 Carry out a procedure to write an equation of a line passing
through two given points.
EA 4.6 Represent linear equations in multiple forms (including
point-slope, slope-intercept, and standard).
EA 5.1 Carry out a procedure to graph a line when given the
equation of the line.
EA 5.3 Carry out a procedure to graph the line with a given slope
and a y-intercept.
EA 5.4 Carry out a procedure to graph the line with a given slope
passing through a given point.
Essential Questions for Learning
Goal 4.8
 What
does point-slope form of
an equation tell me?
Learning Goal 4.8 Notes

Essential knowledge and skills for this unit:





Students must be able to explain how any linear function
in point-slope form can be obtained through
transformations of the parent function y = x.
Students must be able to discuss the effect of changes of
h, k, and m on the equation y – k = m (x – h)
Students must be able to write and graph the equation of
the line in point-slope form, with and without technology
when given a point and a slope, or two points.
Students must be able to convert a given equation to either
of the other two forms.
Students must be able to write the equation in point-slope
form when given the graph of a linear function.
Learning Goal 4.8 Notes



In point-slope form, y-y1 = m(x-x1), m
represents the slope and (x1,y1) is a point on
the line
A general form of the point-slope equation is
y – k = m(x – h) where h is the horizontal
shift and k is the vertical shift of the equation
y = mx.
This means that the point (0,0) on y=mx is
translated to (h,k).
Learning Goal 4.8 Notes

Consider the following example when
thinking about transformations

Y + 3 = 4(x – 5) or Y – (-3) = 4(x – (+5))



Order of operations on the independent variable x
dictates that the parent function y=x is shifted 5
units to the right.
Next, the graph is stretched vertically by a factor
of 4.
Finally, the graph is translated down 3 units.
 If the slope in this example was -4, then the graph
would have been reflected about the line x-axis
before being shifted down 3 units.
Learning Goal 4.8 Notes

Remember that when using point-slope
form either set of coordinate points can
be substituted into the formula.





y-y1 = m(x-x1)
Slope of 2
Coordinates (1,1) and (3,5)
y-1 = 2(x-1), y-1=2x-2, y=2x-1
y-5 = 2(x-3), y-5=2x-6, y=2x-1
Standards for Learning Goal 4.9



EA 1.5 Demonstrate an understanding of algebraic
relationships by using a variety of representations
(including verbal, graphic, numerical, and symbolic).
EA 1.7 Understand how to represent algebraic
relationships by using tools such as handheld computing
devices, spreadsheets, and computer algebra systems
(CASs).
EA 4.1 Carry out a procedure to write and equation of a
line with a given slope and a y-intercept. (Extension)
Essential Questions for Learning
Goal 4.9
 How
do I know if a point is a
solution to a two variable
inequality?
Learning Goal 4.9 Notes

In order to graph an inequality in two
variables, the inequality should first be
written in slope-intercept form.
When you graph an
inequality, you are
forming two half-planes
whose boundary is the
line that corresponds to
the inequality.
Learning Goal 4.9 Notes

When an inequality involves a > or <, graph
a dotted line to show that the points on the
boundary line are not included in the
solution set.
Learning Goal 4.9 Notes

When an inequality involves ≤ or ≥, graph a
solid line to show that the points on the
boundary line are included in the solution
set.
Learning Goal 4.9 Notes

To determine which halfplane should be shaded,
test a point not on the
boundary line by
substituting the
coordinates of x and y into
the inequality.
Shade the half-plane that contains the point
that makes the inequality true.
Learning Goal 4.9 Notes


All of the points (ordered
pairs) in the half-plane
will make the inequality
true and constitute the
solution set of the
inequality.
The origin, (0,0), is the most convenient point
to test, as long as the boundary line does not
go contain the origin.
Essential Knowledge for Learning
Goal 4.9



Students should be able to graph and shade an
inequality in two variables with and without
technology.
Students should be able to write the
corresponding inequality when given a graph.
Students should be able to test to determine
whether it is a solution to a given inequality
when given a point.
Standards for Learning Goal 4.10




EA 1.1 Communicate knowledge of algebraic relationships
by using mathematical terminology appropriately.
EA 1.7 Understand how to represent algebraic
relationships by using tools such as handheld computing
devices, spreadsheets, and computer algebra systems
(CASs).
EA 5.1 Carry out a procedure to graph a line when given
the equation of the line.
EA 5.8 Analyze the equations of two lines to determine
whether the lines are perpendicular or parallel.
Essential Questions for Learning
Goal 4.10
 How
do I write the equations of
parallel and perpendicular lines?
Learning Goal 4.10 Notes
Parallel lines have the same slopes and
different y-intercepts.
 Perpendicular lines have slopes that are
negative reciprocals, m1 · m2 = -1

Learning Goal 4.10 Notes


To write the equation of a linear function
parallel to a given line through a given point,
determine the slope of the given line and use
this slope with the given point in either the
slope-intercept or point-slope form.
To write the equation of a linear function
perpendicular to a given line through a given
point, determine the slope of the given line and
use the negative reciprocal of this lsope with the
given point in either the slope-intercept or pointslope form.
Learning Goal 4.10 Notes

When graphing perpendicular lines on a
graphing calculator, it is necessary to use
the ZoomSquare feature to ensure that the
lines appear perpendicular.
Standards for Learning Goal 4.11




EA 1.2 Connect algebra with other branches of
mathematics.
EA 1.5 Demonstrate an understanding of algebraic
relationships by using a variety of representations
(including verbal, graphic, numerical, and symbolic).
EA 1.7 Understand how to represent algebraic
relationships by using tools such as handheld
computing devices, spreadsheets, and computer
algebra systems (CASs).
EA 5.9 Analyze given information to write a linear
function that models a given problem situation.
Essential Questions for Learning
Goal 4.11
 What
does the solution to a
corresponding one-variable linear
equation mean?
Learning Goal 4.11 Notes

The graphing calculator can be used to find the solutions of
one-variable equations using graphical and tabular methods.

The Graphing Approach:

One way is to enter the left side of the equation in Y1, the right side in
Y2, and graph.
 The x-value of the point of intersection represents the solution to the
original equation.
 This point can be determined by using the Zoom and Trace features of the
calculator.

The Tabular Approach:

Another way is to use the tabular feature of the calculator. The x-value
where Y1 and Y2 are equal is the solution to the equation.
 To zoom-in to more precise values on the table, use the delta-table (Δ TBL)
command to adjust the increment.
 Solutions should be checked by substituting it into the original equation.
Learning Goal 4.11 Notes

The graphing calculator can be used to find the
solutions of one-variable equations using
graphical and tabular methods.

One way is to enter the left side of the equation in Y1, the
right side in Y2, and graph.


The x-value of the point of intersection represents the
solution to the original equation.
This point can be determined by using the Zoom and Trace
features of the calculator.
Learning Goal 4.11 Notes




The solution can also be found by using the calculator’s
Intersect Command.
Another method to finding the solution is to move all terms to one
side of the equation and set the equation equal to zero (0 = ax
+b)
Enter the non-zero side in Y1 (Y1=ax +b) and graph.
The value of x in the linear function where y = 0 is the solution to
the linear equation.


This is the point where the line intersects the x-axis.
The x-value is the zero of the linear function.

The zero of the function is the solution of the equation.
 Note: when reviewing the tabular when look for the x-value when the Y1 is
zero.

The solution of the equation can be found by using the calculator’s
Zero command.
Standards for Learning Goal 4.12



EA 1.4 Judge the reasonableness of
mathematical solutions.
EA 1.5 Demonstrate an understanding of
algebraic relationships by using a variety of
representations (including verbal, graphic,
numerical, and symbolic).
EA 5.10 Analyze given information to determine
the domain and range of a linear function in a
problem situation.
Essential Questions for Learning
Goal 4.12
 Are
the domain values reasonable for
the given situation?
Learning Goal 4.12 Notes

The domain of linear functions is the set of all
real numbers, in many real-world situations
the domain is limited to positive real numbers
or whole numbers.
Standards for Learning Goal 4.13




EA 1.1 Communicate a knowledge of algebraic
relationships by using mathematical terminology
appropriately.
EA 1.2 Connect algebra with other branches of
mathematics.
EA 1.3 Apply algebraic methods to solve problems in realworld contexts.
EA 1.5 Demonstrate an understanding of algebraic
relationships by using a variety of
representations (including verbal, graphic, numerical, and
symbolic).
Standards for Learning Goal 4.13
(Continued)



EA 1.7 Understand how to represent algebraic
relationships by using tools such as handheld
computing devices, spreadsheets, and computer
algebra systems (CASs).
EA 4.4 Use a procedure to write an equation of a
trend line from a given scatterplot.
EA 4.5 Analyze a scatterplot to make predictions.
Essential Questions for Learning
Goal 4.13
 How
do I use a line of best fit?
Learning Goal 4.13 Notes




A scatter plot shows the relationship between two
quantitative variables.
It gives a visual impression of how strongly x-values
and y-values are related.
Scatter plots alone are not sufficient to make precise
statements and draw conclusions from data.
A correlation coefficient is a numerical assessment
of the strength of the relationship between x and y
values in a set of ordered pairs; it ranges from -1 to 1.
Learning Goal 4.13 Notes



A strong linear relationship is indicated if the
absolute value of the correlation coefficient (r)
is close to 1.
If the value of (r) is close to zero, the
relationship is not linear.
The value of correlation coefficient may
appear on the calculator screen when linear
regression is performed.
Learning Goal 4.13 Notes

The association between two variables may be
described as positive correlation, negative
correlation, or no correlation.



Positive Correlation among data means that an increase in x
is associated with an increase in y.
Negative Correlation among data means that as x increases,
y tends to decrease.
No Correlation among data means that it is indicative of no
association between x and y; there is no tendency for y to
either increase or decrease as x increases.
Learning Goal 4.13 Notes



Both scatter plots and best-fit lines can be used to predict
relationships.
The closely the points in the scatter plot “hug” the line, the more
confident you are about the prediction and the stronger the
correlation.
A best-fit line can only be approximated by using technology.


When using the calculator to find the best-fit – use the linear
regression command. Either option (a + bx and Ax +b) will yield the
least-squares regression line.
An alternate approach to writing the equation of a best-fit line
computes the slope as the ratio of average consecutive
differences between y-coordinates as compared to differences in
x-coordinates of the data values.
Learning Goal 4.13 Notes
A strong correlation indicates that two sets of data are
related; however, it does not tell you why the two sets
are related, or if changes in one set of data are
causing the changes in the other set of data.
 A best-fit line may be used to make predictions that
are reasonable.

When making predictions – use input values that are
reasonable, or close to the x values in the data being studied.

For example, when predicting height from age, there are
limitations on both age and height that should be considered.
Work Cited



Carter, John A., et. al. Glencoe
Mathematics Algebra I. New
York: Glencoe/McGrawHill, 2003.
Greenville County Schools Math
Curriculum Guide
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