1. 2.2 Power Functions With Modeling

2. Warm-Up 2
Evaluate
f(-3)
f(-2)
f(2)
f(4)

3. Power Functions with Modeling
Any function that can be written in the form f(x) = kxn, where k and n are nonzero constants, is a power function.
The constant n is the power, and k is the constant of variation, or constant of proportion.
Note: We say f(x) varies as the nth power of x, or f(x) is proportional to the nth power of x.

4. Common Formulas which are power functions
Power Functions with Modeling
Common Formulas which are power functions
Name
Formula
Power
Constant of Variation
Circumference
C = 2πr 1 2π
Area of a circle
A = πr2 2 π
Force of gravity
F = k d2 -2 k
Boyle’s Law
V= k P -1
Circumference of a circle varies directly as its radius. The area enclosed by a circle is directly proportional to the square of its radius. The force of gravity acting on an object is inversely proportional to the squar eo fthe distance from the object to the center of the Earth. The volume of an enclosed gas (at a constant temperature) variables inversely as the applied pressure.

5. Power Functions with Modeling
Direct variation
Two quantities are vary directly if:
As one quantity increases, the other increases, or
As one quantity decreases, the other decreases
In other words, quantities that are directly related must do the SAME things (both increase or both decrease)
Power functions f(x) = kxn that are direct variations have positive powers of n.

6. Inverse variation Two quantities vary inversely if:
As one quantity increases, the other decreases or
As one quantity decreases, the other increases.
In other words, quantities that are inversely related must do OPPOSITE things (one increases while the other decreases).
Power functions f(x) = kxn that are inverse variations have negative powers of n.

7. Comparing Graphs of Monomial Functions
A monomial function is a single-term polynomial function.
Any function that can be written as f(x) = kxa or f(x)=k is a monomial
Typical monomial functions: x, x2, x3

8. Graphs of Monomial Functions
F(x) = xn is
even if n is even
odd if n is odd

9. Determining Power Functions from Data
r2, the coefficient of determination, can be used to determine the appropriateness of a non-linear model.
0 < r2 < 1
If r2 ≈ 1, then the model is more appropriate and is a good representation for the data
If r2 ≈ 0, then the model is less appropriate and is not a good representation for the data.

10. Light Intensity Data for a 100-W Light Bulb
Matthew and Olivia gathered the data shown in the table below from a 100-watt light bulb and a CBL unit with a light-intensity probe.
Find a power function to model this data.
Use the regression model to predict the light intensity at distance of 3.4m.
Light Intensity Data for a 100-W Light Bulb
Distance
(m)
Intensity
(W/m2)
1.0
1.5
2.0
2.5
3.0
7.95
3.53
2.01
1.27
0.90