CCII
Power Models – Day One
Name ____________________________________ #_______
The most common forms of direct and inverse variation can be expressed by the equations y = kx and 𝒚 =
𝒌
𝒙
. But there are other important examples of direct and inverse variation—expressed with rules in the
𝒌
form y = kxr and 𝒚 = 𝒙𝒓. Because the pattern relating x and y is determined by the exponent or power r in
each case, these relationships are called power functions.
Modeling Sound and Light Intensity
The intensity of sound from a voice or of light from a flashlight is related to the distance from the source
to the receiving ear or surface. The more distant the source, the lower the sound or light intensity at the
receiving end.
(1) The following graphs show three possible patterns for functions relating sound or light intensity to
distance. The graphs show distance from source to receiver as the independent variable and sound or
light intensity as the dependent variable.
(a) How would you describe the patterns of change in sound or light intensity shown by each graph?
Graph I: ________________________________________________Graph II: __________________________________________________
Graph III: ______________________________________________
(b) Which graph do you believe is most likely the most accurate model for the relationship between
sound or light intensity and distance from the source? Make your most educated guess and explain the
reasoning behind your choice.
(2) Consider what would happen if you were to enter a dark room and shine a small flashlight directly at
a flat surface like a wall. The flashlight will create a circle of light on the wall.
(a) Complete the table below that contains measurements of light circle diameter for one flashlight that
has been held at several distances from a wall. (Distance and diameter are measured in feet. Express
area in terms of π.)
Light Circle Measurements
Distance from Light Source, x
1
2
3
4
5
6
Diameter of Light Circle, d
2
4
6
8
10
12
Radius of Light Circle, r
Area of Light Circle, A
(b) Use the information and patterns from the table to write equations to represent:
(i) the diameter of light circle as a function of distance to 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) Describe the relationships of the variables by completing sentences in the form: “The variable
________________ is _____________ proportional to ____________________ with constant of proportionality _______.”
(i)
(ii)
(iii)
(d) Light energy is measured in a unit called lumens. The intensity of light is measure in lumens per unit
of area. As the light circle of a flashlight or lamp increases in size, the intensity of light decreases.
To explore how that decrease in light intensity is related to distance from source to target, suppose that
the flashlight that gave the values in the previous table produces 160 lumens of light energy. Use the area
data from the previous table to complete the table below relating light intensity I to distance x.
Distance from Light, x
1
2
Area of Light Circle, A
π
4π
160
𝜋
160
4𝜋
Light Intensity, I
3
4
5
(e) Write an equation that shows light intensity I as a function of distance x from source to surface.
(f) Enter the equation you created in part (e) into Y1 on your calculator. (Note: Be sure to put
parentheses around the denominator when entering in the calculator.)
(a) Look at a graph of the equation. Which graph from Q1 matches the graph on your calculator? _________
Was your prediction in Q1 correct? ___________
(b) Describe the rate of change relationship between the two variables.
(c) What does the pattern of change tell us about the effective range of a flashlight?