Transcript Examples of Direct Variation

Direct Variation
What is it and how do I know when I see it?
What is a Direct Variation or a Direct
Proportion?
 Used
to compare two values and
to see how one value affects the
other value
 When 1 quantity always changes
by the same factor (the
constant) as another, the 2
quantities are in direct
proportion. The 2 quantities
always have equivalent ratios
How do we find know if we have a
direct variation?
 You
can look at:
1. EQUATIONS
2. TABLES
3. GRAPHS to see how two
values affect one another
Direct Variation Equations:
•
Y varies directly as x means that y = kx
where k is the constant of variation.
•
Another way of writing this is k =
y
x
•
X is the independent variable
•
Y is the dependent variable
•
K is the constant of proportionality
Examples of Direct Variation
Equations (y = kx)…
y
y
y
y
y
y
=
=
=
=
=
=
4x
x
2x
2.5x
⅝x
0.75x
k
k
k
k
k
k
=
=
=
=
=
=
4
1
2
2.5
⅝
0.75
Direct Variation & Tables of Values
You can make a table of values for “x”
and “y” and see how the values
behave. You could have a direct
variation if…
 As
“x” increases in value, “y” also
increases in value
OR
 As “x” decreases in value, “y” also
decreases in value
Examples of Direct Variation:
X
6
7
8
Y
12
14
16
Note: As “x” increases,
6,7,8
“y” also increases.
12, 14, 16
What is the constant of variation of the table above?
Start with the direct variation equation: y = kx
Pick one pair of x and y values and substitute into the equation
12 = k · 6
(this is a one-step equation, so solve for k)
12/6 = k →
k=2
Now you can write the equation for this direct variation: y = 2x
Examples of Direct Variation:
Note: X decreases,
X
30
15
9
Y
10
5
3
30, 15, 9
And Y decreases.
10, 5, 3
What is the constant of variation of the table above?
Start with the direct variation equation: y = kx
Pick one pair of x and y values and substitute into the equation
10 = k · 30 (this is a one-step equation, so solve for k)
10/30 = k → (simplify 10/30) → k = ⅓
Now you can write the equation for this direct variation: y = ⅓ · x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
Yes!
X
4
8
12
18
Y
6
12
18
27
y = kx
•Pick an x & y pair and
substitute into the direct
variation equation to solve for k.
•Remember the constant must
hold true for every x,y pair
6=k·4
k = 6/4 = 3/2 = 1 ½
Therefore the equation for this
table is: y = 1 ½ · x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
Yes!
X
10
6
4
2
Y
25
15
10
5
y = kx
* Pick an x & y pair and
substitute into the direct
variation equation & find k.
25 = k · 10
25/10 = k
5/2 or 2 ½ = k
* Remember the constant must
hold true for every x,y pair.
Therefore the equation for this
table is:
y=2½·x
Is this a direct variation? If yes, give the
constant of variation (k) and the equation.
No!
X
15
3
1
2
Y
5
26
75
150
If you look at the values in
the table, you should notice
as “x” decreases, “y”
increases, so you know you
CANNOT have a direct
variation!
Also, there is no constant of
proportionality. There is not
one number you multiply by
x to get y for each pair in the
table.
Which is the equation that describes the
following table of values?
1.
2.
3.
4.
y = -2x
y = 2x
y= ½x
xy = 200
X
10
2
12
20
Y
5
1
6
10
Answer
Now
Using Direct Variation to find unknowns (y = kx)
Given y varies directly with x, and y = 28 when x=7,
Find x when y = 52.
HOW???
2 step process
1. Find the constant of variation
X
Y
y = kx →
7
28
?
52
28 = k · 7
(divide both sides by 7)
k=4
2. Use y = kx. Find the unknown (x).
52= 4x or 52/4 = x
x= 13
Therefore:
X =13 when Y=52
Using Direct Variation to find unknowns (y = kx)
Given that y varies directly with x, and y = 3 when x=9,
Find y when x = 40.5.
HOW???
2 step process
X
Y
3=k·9
9
3
(divide both sides by 9)
40.5
?
1. Find the constant of variation.
Y = kx →
K = 3/9 = 1/3
2. Use y = kx. Find the unknown (x).
y= (1/3)40.5
y= 13.5
Therefore:
X =40.5 when
Y=13.5
Real World Examples of Direct
Variation Situations…
The more time I drive at a constant rate,
the more miles I go.
 If I increase a recipe for more people, the
more of an ingredient I need.
 The more hours I work, the more money I
make.
 The more CD’s I purchase, the more
money it costs.
 The less cheese I buy at the deli, the less
money I pay.

Using Direct Variation to solve word problems
Problem:
A car uses 8 gallons of
gasoline to travel 290
miles. How much
gasoline will the car use
to travel 400 miles?
Step Two: Find the constant
of variation and equation:
y = kx → 290 = k · 8
290/8 = k
y = 36.25 x
Step One: Find points in table
X (gas) Y (miles)
8
290
?
400
Step Three: Use the equation
to find the unknown.
400 =36.25x
400 =36.25x
36.25 36.25
or x = 11.03
Using Direct Variation to solve word problems
Problem:
Step One: Find points in table.
Julio wages vary
directly as the number
of hours that he works.
If his wages for 5 hours
are $29.75, how much
will they be for 30 hours
Step Two: Find the constant of
variation.
y = kx → 29.75 = k · 5
k = 5.95
X(hours) Y(wages)
5
29.75
30
?
Step Three: Use the equation
to find the unknown.
y = kx
y = 5.95 ·30
y = 178.50
Direct Variation and Its Graph
Characteristics of Direct Proportion
Graph…
 The
graph will always go through the
ORIGIN (point 0,0) on the coordinate
plane) & will be in Quadrants I and
III only (for this year we’re only
required to graph in Quadrant I – all
positive numbers)
 The graph will always be a straight
line
 As the “x” values increase, the “y”
value will also increase
Independent VS. Dependent
 The
x-axis is the independent
variable; this means it does NOT
depend on the y value
 The y-axis is the dependent
variable; this means it DOES
depend on the x variable for its
value.
Example…

You went on a hiking trip, and you
graphed your distances at various times
throughout your trip. As the time you
hiked increased, the distance traveled also
increased. The distance depends on the
time; therefore, the distance is the
dependent variable and it is graphed on
the y-axis (vertical). The time is
independent (time still goes on whether or
not the distance changes), so it is graphed
on the x-axis (horizontal).
Tell if the following graph is a Direct Variation or not.
No
No
No
No
Tell if the following graph is a Direct Variation or not.
No
Yes
Yes
No