Objective: Students will be able to identify direct and inverse variation and solve direct and inverse variation problems as evidenced by teacher monitoring. Do Now: Check your answers to the HW.
What is Direct Variation?
A direct variation between two variables, x and y,
is a relationship that is expressed as:
y = kx
where k is the constant of
variation/proportionality
“As the value of x increases/decreases, the value of y increases/decreases by the same scale factor.”
What does this mean?
y = 2x Same x y Scale Factor! y = 2x 1 2 x x ×2 ×2 2 4 Same y ×1.5 ×1.5 Scale Factor! =2 3 6 x “y varies directly with x”
k-constant of variation
or
“y is directly proportional to x”
Is this a Direct Variation?
“Does y vary directly with x?”
Example 1:
×2 ×1.5 ×
4 3
Example 2:
x y x y 1 2 ×2.5 ×2 1 4 ×2 2 8 2 5 ×1.6 ×1.5 ×1.5 3 12 3 8 × × × 4 16 4 11 11 8
No!
y = 3x − 2
What is the constant of variation?
4 3
4 3
Yes!
y = 4x x x
Joint Direct Variation
Example 1: The variable y varies directly with x. If y = 75
when x = 10, then find y when x = 16.
Varies by some constant of variation!
If y = kx, then
y k= . x
y = 7.5x
y1 y2 Cross 75 y2 = → = Multiply! x1 x2 10 16 10y = 1200 Check 75 120 10 10 = 10 16 y = 120 7.5 = 7.5
Joint Direct Variation
Example 2: In a factory, the profit, P, is directly
proportional to the inventory, I. If P = 100
when I = 20, then find P when I = 50.
P1 P2 100 P = → = I1 I 2 20 50
Cross Multiply!
20P = 5000 20 P = 5I 100 = 250 20 20 50 P = $250 Check
k
5=5
What is Inverse Variation?
An inverse variation between two variables, x and y, is a relationship that is expressed as:
k y= x
where k is the constant of
variation/proportionality
“As the value of x increases/decreases, the value of y does the opposite by the reciprocal scale factor of x.”
What does this mean?
6 y= x Reciprocal 6 x y Scale Factors! ( x) y = x ( x) 1 6 × ×2 Reciprocal 2 3 Scale Factors! xy = 6 × × 3 2 k-constant of variation
1 2
3 2
2 3
“y varies inversely with x”
or
“y is inversely proportional to x”
Is this an Inverse Variation?
“Does y vary inversely with x?”
Example 1:
x y Example 2:
x ×2 ×
3 2
× 43 What is the constant of variation?
1 2 3 4
24 12 8 6
×
1 2
×2
×
2 3
× 23
×
3 4
×
4 3
y 1 37 2 19 3 13 4 10
× 19 37 13 × 19
× 10 13
Yes!
No!
24 ( x) y = x ( x)
36 y= +1 x
Joint Inverse Variation
Example 1: The variable y varies inversely with x. If
y = 168 when x = 24, then find y when x = 30.
Varies by some constant of variation!
k If y = , then x xy = k.
x1 y1 = x2 y2
(24 ) (168) = 30y 4032 = 30y 30 30 y = 134.4
Joint Inverse Variation
Example 1: The variable y varies inversely with x. If
y = 168 when x = 24, then find y when x = 30.
x1 y1 = x2 y2
Check
(24)(168) = (30)(134.4) 24 168 = 30y ( )( )
4032 = 30y 30 30 y = 134.4
4032 = 4032
k
4032 y= x
Joint Inverse Variation
Example 2: The time, t, it takes to drive a certain distance
is inversely proportional to the rate of speed, r.
If Elizabeth drives 5 hrs at 55 mph, how long
would it take her to make the same trip if she
were to drive at 50 mph?
Check r1t1 = r2t 2 (55)(5) = (50)(5.5) (55) (5) = 50t 275 = 275 275 = 50t k 275 50 50 t = 5.5 hrs = t r
Regents Question
Example 3: The points (2, 3), (4, ¾ ), and (6, d) lie on the
graph of a function. If y is inversely
proportional to the square of x, what is the
value of d.
2 1 1
2 2 2 2
x1 y1 = x2 y2 → x y = x y
Check
(2
2
) ( 3) = (6 12 = 12
k
(2 ) ( 3) = 6 d 2
2
12 y= 2 x
)( ) 1 3
12 = 36d 36 36 1 =d 3
