Transcript inverse variation

12-8 Inverse Variation
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Pre-Algebra
12-8 Inverse Variation
Pre-Algebra HOMEWORK
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Pre-Algebra
12-8 Inverse Variation
Students will be able to solve sequences and represent
functions by completing the following assignments.
•
•
•
•
Learn to find terms in an arithmetic sequence.
Learn to find terms in a geometric sequence.
Learn to find patterns in sequences.
Learn to represent functions with tables, graphs, or
equations.
• Learn to identify linear functions.
• Learn to recognize inverse variation by graphing
tables of data.
Pre-Algebra
12-8 Inverse Variation
Today’s Learning Goal Assignment
Learn to recognize
inverse variation
by graphing tables
of data.
Pre-Algebra
12-8
12-8Inverse
InverseVariation
Variation
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
Pre-Algebra
12-8 Inverse Variation
Warm Up
Find f(–4), f(0), and f(3) for each
quadratic function.
1. f(x) = x2 + 4
20, 4, 13
9
2. f(x) = 1 x2
4, 0, 4
4
3. f(x) = 2x2 – x + 3 39, 3, 18
Pre-Algebra
12-8 Inverse Variation
Problem of the Day
Use the digits 1–8 to fill in 3 pairs of
values in the table of a direct variation
function. Use each digit exactly once.
The 2 and 3 have already been used.
8
1
Pre-Algebra
56
4
7
12-8 Inverse Variation
Vocabulary
inverse variation
Pre-Algebra
12-8 Inverse Variation
INVERSE VARIATION
Words
An inverse variation
is a relationship in
which one variable
quantity increases as
another variable
quantity decreases.
The product of the
variables is a constant.
Pre-Algebra
Numbers
120
y=
x
xy = 120
Algebra
y=
k
x
xy = k
12-8 Inverse Variation
Additional Example 1A: Identify Inverse Variation
Tell whether the relationship is an inverse
variation.
A. The table shows how 24 cookies can be
divided equally among different numbers of
students.
Number of Students
Number of Cookies
2
3
4
6
8
12
8
6
4
3
2(12) = 24; 3(8) = 24; 4(6) = 24; 6(4) = 24; 8(3) = 24
xy = 24
The product is always the same.
The relationship is an inverse variation: y = 24
x .
Pre-Algebra
12-8 Inverse Variation
Try This: Example 1A
Tell whether the relationship is an inverse
variation.
A.
x
0
0
0
0
0
y
2
3
4
5
6
0(2) = 0; 0(3) = 0; 0(4) = 0; 0(5) = 0; 0(6) = 0
xy = 0
The product is always the same.
The relationship is an inverse variation: y = x0 .
Pre-Algebra
12-8 Inverse Variation
Additional Example 1B: Identify Inverse Variation
Tell whether each relationship is an inverse
variation.
B. The table shows the number of cookies that
have been baked at different times.
Number of Students
12
24
36
48
60
Time (min)
15
30
45
60
75
12(15) = 180; 24(30) = 720
The product is not
always the same.
The relationship is not an inverse variation.
Pre-Algebra
12-8 Inverse Variation
Try This: Example 1B
Tell whether the relationship is an inverse
variation.
B.
x
y
2
4
4
2
8
1
2(4) = 8; 2(6) = 12
1
8
2
6
The product is not
always the same.
The relationship is not an inverse variation.
Pre-Algebra
12-8 Inverse Variation
Additional Example 2A: Graphing Inverse Variations
Graph the inverse variation function.
A. f(x) = 4
x
y
x
–4 –1
–2 –2
–1 –4
–
1
2
1
2
8
1
4
2
2
1
4
Pre-Algebra
–8
12-8 Inverse Variation
Try This: Example 2A
Graph the inverse variation function.
A. f(x) = – 4
x
x
y
–4 1
–2 2
–1 4
–
1
2
1
2
–8
1
–4
2
–2
–1
4
Pre-Algebra
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12-8 Inverse Variation
Additional Example 2B: Graphing Inverse Variations
Graph the inverse variation function.
x
y
B. f(x) = –1
x
–3 1
–2
–1
1
–
2
1
2
1
2
3
Pre-Algebra
3
1
2
1
2
–2
–1
–1
2
–1
3
12-8 Inverse Variation
Try This: Example 2B
Graph the inverse variation function.
B. f(x) = 8
x
y
x
–8 –1
–4 –2
–2 –4
–1 –8
1
8
2
4
4
2
1
8
Pre-Algebra
12-8 Inverse Variation
Additional Example 3: Application
As the pressure on the gas in a balloon
changes, the volume of the gas changes. Find
the inverse variation function and use it to
find the resulting volume when the pressure is
30 lb/in2.
Volume of Gas by Pressure on Gas
5
10
15
20
Pressure (lb/in2)
300
150
100
75
Volume (in3)
You can see from the table that xy = 5(300) = 1500,
so y = 1500 .
x
If the pressure on the gas is 30 lb/in2, then the
volume of the gas will be y = 1500 ÷ 30 = 50 in3.
Pre-Algebra
12-8 Inverse Variation
Try This: Example 3
An eighth grade class is renting a bus for a
field trip. The more students participating, the
less each student will have to pay. Find the
inverse variation function, and use it to find
the amount of money each student will have
to pay if 50 students participate.
Number of Students by Cost per Student
10
20
25
40
Students
20
10
8
5
Cost per student
You can see from the table that xy = 10(20) = 200, so
y = 200 .
x
If 50 students go on the field trip, the price per
student will be y = 200  50 = $4.
Pre-Algebra
12-8 Inverse Variation
Lesson Quiz: Part 1
Tell whether each relationship is an inverse
variation.
1.
yes
2.
no
Pre-Algebra
12-8 Inverse Variation
Lesson Quiz: Part 2
1 .
3. Graph the inverse variation function f(x) = 4x
Pre-Algebra