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Unit 4 Seminar
GRAPHS
• 4.1 Variation
• 4.2 Linear Inequalities
• 4.3 Graphing Linear Equations
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4.1
Variation
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Direct Variation
• If a variable y varies directly with
a variable x, then y = kx, where k is
the constant of proportionality.
• If x increases, then y increases.
• If x decreases, then y decreases.
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EXAMPLE: The amount of interest earned on an investment, I,
varies directly as the interest rate, r. If the interest earned is
$50 when the interest rate is 5%, find the amount of interest
earned when the interest rate is 7%.
1) I = kr
2) $50 = 0.05k
1000 = k
Write the variation equation, I varies directly r
Find k by substituting the given variables
3) k = 1000, r = 7%
I = 1000r
Use k to determine I when r is given
I = 1000(0.07)
I = 70
The amount of interest earned is $70.
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Inverse Variation
• If a variable y varies inversely with a
k
variable x, then y = , where k is
x
the constant of proportionality.
• If x increases, then y decreases.
• If x decreases, then y increases.
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EXAMPLE: Suppose y varies inversely as x.
If y = 12 when x = 18, find y when x = 21.
k
1)
y
x
k
12 
18
216  k
2)
k
y
x
216
y
21
y  10.3
Write the variation equation, y varies inversely as x
Find k by substituting the given variables
Use k to determine y when x is given
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Joint Variation
• The general form of a joint variation,
where y varies directly as x and z, is
y = kxz, where k is the constant of
proportionality.
• As x and z increase, y increases.
• As x and z decrease, y decreases.
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EXAMPLE: The area, A, of a triangle varies
jointly as its base, b, and height, h. If the area
of a triangle is 48 in2 when its base is 12 in. and
its height is 8 in., find the area of a triangle
whose base is 15 in. and whose height is 20 in.
A  kbh
48  k(12)(8)
48  k(96)
48 1
k

96 2
A varies jointly
as b and h
Find k using the
given values
A  kbh
1
A  (15)(20)
2
A  150 in.2
Using k, find A
when b and h
are given.
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EXAMPLE: A varies jointly as B and C and inversely as
the square of D. If A = 1 when B = 9, C = 4, and D = 6,
find A when B = 8, C = 12, and D = 5.
• Find the constant of
proportionality.
kBC
A 2
D
k (9)(4)
1
62
36k
1
36
1 k
• Now find A.
kBC
A 2
D
(1)(8)(12)
A
52
96
A
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A  3.84
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EVERYONE: pg 151 #44
Making an Ice Cube The time, t,
for an ice cube to melt is inversely
proportional to the temperature, T,
of the water in which the ice cube is
placed. If it takes an ice cube 2
minutes to melt in 75 degrees F
water, how long will it take an ice
cube of the same size to melt in 80
degrees F of water?
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ANSWER to:
pg 151 #44
t = k/T
t = k/T
2 = k/75
t = 150/80
150 = k
t = 1.875 min
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4.2
Linear Inequalities
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Inequality signs
•
< is less than
(example: 3 < 5)
•
≤ is less than or equal to
•
> is greater than (example: 5 > 3)
•
≥ is greater than or equal to
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Graphing Inequalities
• x<2
The numbers less than to 2 are all the points on the
number line to the left of 2. The open circle at 2
shows that 2 is NOT included in the solution set.
• x≤4
The numbers less than or equal to 4 are all the
points on the number line to the left of 4 and 4
itself. The closed circle at 4 shows that 4 is
included in the solution set.
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Graphing Inequalities
• x>3
The numbers greater than to 3 are all the points on
the number line to the right of 3. The open circle at
3 shows that 3 is NOT included in the solution set.
• x ≥ -5
The numbers greater than or equal to -5 are all the
points on the number line to the right of -5 and -5
itself. The closed circle at -5 shows that -5 is
included in the solution set.
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Example: Solve and graph the solution
• Solve 3x – 8 < 10 and graph the solution set.
3 x  8  10
3 x  8  8  10  8
3 x  18
3 x 18

3
3
x6
• The solution set is all real numbers less than 6.
10 - 3x ≤ 21
10 - 3x ≤ 21
-3x ≤ 11
When dividing by a
negative number, flip
the inequality sign.
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When multiplying by a
negative number, flip
the inequality sign.
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Compound Inequality
• Graph the solution set of the inequality
4 < x  3 where x is a real number
• The solution set consists of all real
numbers between 4 and 3, including the
3 but not the 4.
Get the x alone in the middle…
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Example
A student must have an average (the
mean) on five tests that is greater than
or equal to 85% but less than 92% to
receive a final grade of B. Jamal’s
scores on the first four tests were
98%, 89%, 88%, and 93%.
What range of scores on the fifth test
will give him a B in the course?
Example (continued)
• Let x = Jamal’s score on the fifth test. Then:
98  89  88  93  x
85 
 92
5
368  x
85 
 92
5
5(85)  368  x  92(5)
425  368  x  460
425  368  368  368  x  460  368
57  x  92
• So Jamal will receive a grade of B in the course if his
score on the fifth test is greater than or equal to 57
and less than 92.
4.3
Graphing Linear
Equations
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Rectangular Coordinate System
• The rectangular coordinate system consists of a horizontal axis
called the x-axis and a vertical axis called the y-axis.
• The x- and y-axes intersect at a point called the origin, dividing the
plane into four regions called quadrants.
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• Each point in the plane corresponds to an ordered pair (x, y) of real
numbers.
• The x and y values are called the coordinates of the point.
• The coordinates for the origin are (0, 0).
• The x-coordinate tells how many units the point is to the left or
right of the origin.
• If x is positive, the point is to the right of (0, 0).
• If x is negative, the point is to the left of (0, 0).
• The y-coordinate tells how many units the point is up or down from
the origin.
• If y is positive, the point is above (0, 0).
• If y is negative, the point is below (0, 0).
• Locating a point in the plane is called plotting the point.
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Example:
Plot the point (3, 5).
(3, 5)
1. Move right along the
horizontal x-axis to 3.
2. Move up 5 units until
aligned with the 5 on
the vertical y-axis.
3. Draw a point
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Example: Plot the point (4, –2).
1. Move right along the
horizontal x-axis to 4.
(4, –2)
2.Move down 2 units
until aligned with the
–2 on the vertical yaxis.
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Example: Plot the point (–2, 4).
(–2, 4)
1. Move left along the
horizontal x-axis to
–2.
2. Move up 4 units until
aligned with the 4 on
the vertical y-axis.
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Example: Plot the point (–3, 0).
1. Move left along the
horizontal x-axis to –3.
(–3, 0)
2. The 0 indicates that we
do not move along the
vertical y-axis.
3. The point is plotted on
the x-axis.
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Example: Plot the point (0, –5).
1. The 0 indicates that
we do not move along
the horizontal x-axis.
2. Move down 5 units to
the –5 on the vertical
y-axis.
(0, –5)
3. The point is plotted on
the y-axis.
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Linear Equations
• Linear equations are a type of equation
whose graph is a straight line.
• The solutions of a linear equation are
the ordered pairs (x, y) that make the
equation true.
• The solutions are points that are on the
line.
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Example:
Use a table of values to sketch
the graph of the linear equation 2x - y = 4.
1)
Solve the equation for y.
 2x - y = 4 is equivalent to y = 2x – 4
2)
Select at least 3 values for x and create a
table of values.
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Example: cont’d
3. Plot the points
from the table
and connect
them using a
straight line
with arrow tips
on the ends.
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Intercepts
• The x-intercept of a linear equation
graph is the point where the line crosses
the x-axis.
– An x-intercept has the form (x, 0).
• The y-intercept of a linear equation graph
is the point where the line crosses the yaxis.
– A y-intercept has the form (0, y).
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• The x-intercept of this graph is (2, 0).
• The y-intercept of this graph is (0, -4).
x-intercept (2,0)
y-intercept (0,-4)
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Example: Draw the graph of the linear equation
y = 3x + 6 using the x- and y-intercepts.
• The x-intercept is found by
letting y = 0 and solving for x.
y = 2x + 4
0 = 2x + 4
4 = 2x
2= x
• The y-intercept is found by
letting x = 0 and solving for y.
y = 2x + 4
y = 2(0) + 4
y=4
Example: Graph 3x + 2y = 6
• Find the x-intercept.
3x + 2y = 6
3x + 2(0) = 6
3x = 6
x=2
• Find the y-intercept.
3x + 2y = 6
3(0) + 2y = 6
2y = 6
y=3
Slope
• The slope of a line can be defined as the
steepness of the line.
• If you want to find the slope of a straight
line, choose any two points on the line and
count the number of units in the rise (up or
down) and the run (right) from one point to
the other.
• The slope is the ratio of the rise to the run.
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Slope, cont’d
• For example, you can determine the slope
of an airplane’s flight path shortly after
takeoff, as shown below.
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For
,
you can "rise" up or down
but, you ALWAYS
"run" to the right.
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Example:
Calculate the slope of the line.
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Example: Solution
• From point A to point B
we count up 4 units and
run 7 units.
• The slope is
• Notice that the slope goes
uphill and
is a positive
number
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Example: Calculate the slope of the line.
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Slope, cont’d
• Lines with positive slopes are increasing as they
move from left to right.
• Lines with negative slopes are decreasing as
they move from left to right.
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Slope, cont’d
We can also calculate the slope of a
line, without looking at the graph, if
we know two points on the line.
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Example: Calculate the slope of the line.
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Example: Calculate the slope of the
line that passes through the points
(0, -5) and (1, -3).
Solution: Using (0, -5) and (1, -3),
The slope of the line is 2.
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Example: Calculate the slope of the
line that passes through the points
(2, 3) and (2, -1).
Solution: : Using (2, 3) and (2, -1),
The slope of the line
is undefined.
.
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Example: cont’d
• Solution, cont’d:
If the slope of a
line is undefined,
the graph is a
vertical line.
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Example: Calculate the slope of the
line that passes through the points
(-2, 4) and (3, 4).
Solution: Using (-2, 4) and (3, 4),
The slope of the line is 0.
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Example: cont’d
Solution, cont’d:
If the slope of a line
is 0, the graph is
a horizontal line.
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Summary: Types of Slope
• Positive slope rises
from left to right.
• Negative slope falls
from left to right.
• The slope of a
vertical line is
undefined.
positive
• The slope of a
horizontal line is
zero.
zero
negative
undefined
Graphing Linear Equations Using
Slope Intercept Form
If the general form of a linear equation is
solved for y, the equation is then in
slope-intercept form.
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Example:
It crosses the y-axis at 4,
so we start there:
the slope is
so we
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Example:
It crosses the y-axis at -2,
so we start there:
the slope is 4 which is really
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Horizontal Lines
• A horizontal line has
an equation of the
form y = b.
y = 3
• The equation of the
graph shown is
y = 3
• The slope of a
horizontal line is 0!
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Vertical Lines
• A vertical line has an
equation of the form
x = a.
x = 2
• The equation of the
graph shown is
x = 2
• The slope of a vertical
line is undefined!
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