1. Notes Quiz Turn in notes from Sep. 12th and Sep. 18th
Be sure your NAME is on each set of notes
They should be in the box BEFORE you start the warm-up activity

2. Warm-up (5 min. – No talking)
Power Functions with Modeling
Warm-up (5 min. – No talking)
Find the average rate of change for f(x) = 3x + 5 from
x = -12 and x = 15
Find the average rate of change for g(x) = -x2 + 3 from
x = 7 and x = 9
Describe the type and strength of the
correlation of the scatter plot shown
at the right.
Use g(f(x) to determine if
f(x) = 1 – x and g(x) = 1 – 2x
3x x + 1 are inverse functions.

3. 2.2 Power Functions With Modeling
Sketch power functions in the form f(x) = kxn (where k and a are rational numbers).
Use power functions to model real-life data and use these models to make predictions

4. Power Functions
Any function that can be written in the form f(x) = kxa, where k and a are nonzero constants, is a power function.
The constant a is the power, and k is the constant of variation, or constant of proportion.
In general, if y = f(x) varies as a constant power of, then y is a power function of x.
Note: We say f(x) varies as the ath power of x, of f(x) is proportional to the ath power of x.

5. Common Formulas which are power functions
Power Functions with Modeling
Common Formulas which are power functions
Name
Formula
Power
Constant of Variation
Circumference
C = 2pr 1 2p
Area of a circle
A = pr2 2 p
Force of gravity
F = k/d2 -2 K
Boyle’s Law
V = k/P -1 k
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.

6. Consider this…
What happens to the circumference of a circle as its radius increases?
What happens to the area of a circle as its radius increases?
What happens to the force exerted on an object as the square of the distance from the Earth’s center increases?
What happens to the volume of gas inside an object as the pressure exerted on the object increases?

7. Direct or Inverse? Direct/Inverse Name Formula Power
Constant of Variation
Circumference
C = 2pr
Area of a circle
A = pr2
Force of gravity
F = k/d2
Boyle’s Law
V = k/P

8. Direct or Inverse? Direct/Inverse Name Formula Power
Constant of Variation
Circumference
C = 2pr 1 2p
Area of a circle
A = pr2 2 p
Force of gravity
F = k/d2 -2 k
Boyle’s Law
V = k/P -1
Direct
Direct
Inverse
Inverse

9. Direct variation
Two quantities are vary directly if as one quantity increases, the other increases or as one quantity decreases, the other decreases as well.
In other words, quantities that are directly related must do the SAME things (both increase or both decrease)
Power functions f(x) = kxa that are direct variations have positive powers of a.
i.e. f(x) = x (the square root function)

10. Inverse variation
Two quantities are 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) = kxa that are direct variations have negative powers of a.
i.e. f(x) = 1/x (the reciprocal function)

11. Writing a power function
Ex1 The area A of an equilateral triangle varies directly as the square of the length s of its side. Express the area as a power function in terms of s.
Does anyone know the value of the constant of variation, k?
(3)/4

12. Do Now
Which of the twelve basic functions we discussed are power functions? State the power of each.
Identity f(x) = x
Squaring f(x) = x2
Cubing f(x) = x3
Reciprocal f(x) = 1/x
Square root f(x) = x

13. Ex 2 State the power and constant of variation for the function, graph it, and analyze it. (domain, range, continuity, increasing/decreasing, symmetry, boundedness, local extrema, asymptotes, and end behavior)

14. Comparing Graphs of Monomial Functions
A monomial function is a single-term polynomial function
The power functions of the form f(x) = kxa are all monomials.
Complete the Exploration on p. 183
Which ordered pairs do the functions have in common?

15. Ex 3 Describe how to obtain the graph of g(x) given the graph of f(x) = xn with the same power of n.
g(x) = 1.5x5
g(x) = -2x6

16. Ex 4 State the values of k and a for the function f(x)
Ex 4 State the values of k and a for the function f(x). Describe the portion of the curve that lies in Quadrant I or IV. Determine whether f is even, odd, or undefined for x < 0. Describe the rest of the curve if any. Graph the function to see whether it matches your description.
f(x) = -4x2/3
f(x) = -x-4

17. Determining Power Functions from Data
Recall 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.

18. Light Intensity Data for a 100-W Light Bulb
Ex 5 Velma and Reggie gathered the data shown in the table below from a 100-watt light bulb and a CBL unit with a light-intensity probe.
Draw a scatter plot of the data
Find a regression model.
Is the power close to the a = -2 (the theoretical value)?
Sketch the graph of the equation along with the data points.
Use the regression model to predict the light intensity at distances of 1.7m and 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

19. Tonight’s Assignment
P Ex 3-54 m. of 3, 60-63
Questions?