Unit 1-Functions, Equations and Systems
Lesson 1-Direct and Indirect Variation-Investigation 2
NCES : MBC.A.1.2; MBC.A.3.2; MBC.A.8.5; MBC.A.10
NCSCS: Algebra I 1.03; Algebra II 1.05
Name ________________
Date__________________
(Developed by PHS for use as a resource with Core-Plus Mathematics Course 2, 2nd Edition, Glencoe/ McGraw Hill, © 2008. )
By the end of this investigation you should be able to answer:
What are the patterns of variation that can be modeled well by power functions and what
practical and scientific problems can be solved by use of these functions?
Power Models (page 10)
Power Functions—special examples of direct and indirect variations.
Modeling Sound and Light Intensity
1. 3 graphs relating sound or light intensity to distance:
.
a. Patterns of change in sound or light intensity……
b. Which graph do you think models the relationship between sound or light intensity to distance
the best? Why?
2. Flashlight shining on wall:
a. Complete the table below for the diameter of the light circle of the flashlight that is x distance
from the wall. Remember: radius = ½ diameter
area of circle
b. Write rules for:
i. diameter of light circle as a function of distance from light source
ii. radius of light circle as a function of distance from light source
iii. Area of light circle as a function of distance from the light source
c. Complete the following sentence for diameter, radius and area:
The variable __diameter_____ is _______________ proportional to ___________, with
constant of proportionality ______.
The variable _______________ is _______________ proportional to ___________, with
constant of proportionality ______.
The variable _______________ is _______________ proportional to ___________, with
constant of proportionality ______.
d.
light energy → measured in lumens
light intensity →
As light circle increases in size, the intensity of the light decreases.
Suppose the flashlight produces 160 lumens of light energy. Use the data from a) to fill in the
table below relating light intensity to x:
e. rule for relating light intensity, I, to distance from light, x
(distance, light intensity)
f. i. Which graph from Problem 1 models best the rule from e)?
ii. Describe the pattern of change for e).
Describe the effectiveness of the range of the flashlight.
The Power Function Family (page 12)
3. Explore the shape of the graphs for a direct variation function
of k and r. Start with k = 1 and r = 1,2,3,4,5,6. Then change k to 2, -1,-2.
for different values
Look at the shapes of :
y = 1x1
y = 1x2
y = 1x3
y = 1x4
y = 1x5
y = 1x6
then look at
y = 2x1
y = 2x2
etc.
then
y = -1x1
etc.
then
y = -2x1
etc.
a. Describe the patterns you found when doing the above. Can you explain why they make
sense?
4. Explore the shape of the graphs for a direct variation function
for different values of k and r. Start with k = 1 and r = 1,2,3,4,5,6. Then change k to 2, -1,-2.
(Follow #3’s instructions.)
a. Describe the patterns you found when doing the above. Can you explain why they make
sense?
Time (sec)
Modeling Roll-Time Data Patterns (page 13)
Scientific theory
be
predict that this function should
Platform Height (ft)
“Time is inversely proportional to the square root of platform height, with constant of
proportionality 2.”
5.
Use data from Investigation 1 and the function
to complete the table.
a. Plot the experimental (H,T) values and the function
on the graph.
b. Why will the theoretical (predicted) and experimental (actual) results be somewhat different?
6. a. Use your calculator to find a power model for the data (distance,area) from Problem 2a.
Is this similar to what you derived earlier?
b. Do the same with (distance, intensity) from Problem 2d.
Is this similar to what you derived earlier?
c. Do the same with (platform height, experimental roll time) from Problem 5.
Summarize the Math (page 14)
Sketch a graph and give a brief description for the following:
a. direct variation power function
i. r > 0 and even, k >0
ii. r > 0 and odd, k > 0
b.
i. r > 0 and even, k < 0
ii. r > 0 and odd, k < 0
c. inverse variation power function
i. r > 0 and even, k > 0
ii. r > 0 and odd, k > 0
d.
i. r > 0 and even, k < 0
ii. r > 0 and odd, k < 0
e. What types of graphs will you get for direct and inverse variation when r is 0.5 or -0.5?
Check Your Understanding (page 15)
Match each function rule to its graph and write a sentence beside each in the form, “y is
____________ proportional to ____________ with constant of proportionality ______.”
a. y = 0.5x2
Graph ______
b.
Graph ______
c. y = x3
Graph ______
d.
Graph______
e.
Graph______
f. y = (0.5x)
Graph______
On Your Own (page15)
#4, 5, 6, 7, 11, 16, 17, 19, 20
Review # 26, 27