Transcript 2-1 Power and Radical Functions

2-1 Power and Radical Functions
Part 1
Chapter 2
Power, Polynomial, and Rational Functions
Objectives
 You have already analyzed parent functions and their families
of graphs.
Now we will be
 Graphing and analyzing power functions,
 Graphing and analyzing radical functions, and
 (Solving radical equations next class)
1. Power functions
 Power functions previously
studied are
f ( x)  x
 A power function is any
function of the form
2
f ( x )  ax
n
where a and n are nonzero
constant real numbers.
and
f ( x)  x
3
End Behavior: using f(x)= axn
When a > 0
When n is even
When n is odd
When a < 0
As x  –∞,
As x  –∞,
As x  ∞,
As x  ∞,
As x  –∞,
As x  –∞,
As x  ∞,
As x  ∞,
We should be able to also identify domain, range,
intercepts, continuity, and where the function is increasing
and decreasing.
Example 1a:
f ( x) 
1
x
6
2
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
Example 1b:
f ( x)   x
5
 Graph and analyze each function. Describe the domain,
range, intercepts, end behavior, continuity, and where the
function is increasing or decreasing.
2. Functions with Negative Exponents
 Recall that f ( x ) 
1
or x
1
is undefined at x = 0.
x
2
 Similarly, f ( x )  x
and f ( x )  x  3 are undefined at x = 0.
 Because power functions can be undefined when n < 0 (negative
exponents), these functions will have discontinuities.
Example 2a:
f ( x)  2 x
4
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
Example 2b: f ( x )  2 x
3
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
3. Functions with Rational Exponents
1
 Recall that f ( x )  x n is a root function.
p
 f ( x )  x n indicates the nth root of xp
p
 If n is an even integer, then the domain must be restricted to
f (x)  x n
nonnegative values.
3
Example 3a:
f (x)  2 x 4
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
5
Example 3b:
f ( x )  10 x 3
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
4. Graphing Radical Functions
 An expression with rational exponents can be written in
radical form.
 Exponential form
vs. Radical form
p
f (x)  x n
f ( x) 
n
x
p
Example 4a:
f ( x)  5 2 x
3
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
Example 4b:
f ( x) 
1
5
3x  4
2
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
Assignment: p. 92
 1, 5, 9, 13,19, 23, 27, 35, 37
2-1 Power and Radical Functions
Part 2
Chapter 2
Power, Polynomial, and Rational Functions
Warm up
 Graph and analyze the function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function
is increasing or decreasing.
5
1.
g ( x) 
2.
h( x)  3 x
x
8
8
7
3. f ( x )  3 6  3 x
Answer: 1.
g ( x) 
5
8
x
8
Answer: 2.
h( x)  3 x
7
Answer: 3.
f ( x)  3 6  3x
Questions on the homework?
Objectives:
Learn how to use a graphing calculator to find a power
function to model a set of data.
2. Solve radical equations.
1.
3.
Homework quiz over last night’s homework.
4. Power Regression
 The table shows the braking distance in feet at several speeds
in miles per hour for a specific car on a dry, well-paved
roadway.
Speed
10
20
30
40
50
60
70
Distance
4.2
16.7
37.6
66.9
104.5
150.5
204.9
 If we wanted to find a model to represent this relationship, a
graphing calculator should make this an easy task.
 Write these steps down in your notes to refer to later, when
you need to do another model.
Speed
10
20
30
40
50
60
70
Distance
4.2
16.7
37.6
66.9
104.5
150.5
204.9
1. Enter the data in L1 (speed) and L2 (distance)
(To access, press STAT, then EDIT. Clear the old lists but DO NOT
DEL(ete).
2. Create a scatter plot of the data
(2nd STATPLOT, Select Plot 1, Turn it on [if it is off]. Select a scatter
plot, and the appropriate lists. Press ZOOM 9)
3. If the plot appears to be a power function, use the use the STAT, CALC,
PwrReg function.
(To place this equation in the Y= menu, go to Y1=, then VARS,
Statistics, EQ, RegEQ)
4. Check your scatter plot again. The regression line should appear on the
screen with the data points.
5. Use the equation to make a prediction of the braking distance of a car
going 80 miles per hour. A table of values starting at 79 would work
just fine.
Solving Radical Equations
First, isolate the radical expression.
Then, raise each side of the equation to a power equal to the
index of the radical to eliminate the radical.
Note: This process sometimes produces extraneous solutions,
which are solutions which do not satisfy the original
equation.
Always check your solutions in the original equation to
determine if extraneous solutions exist.
Example 1.
2x 
28 x  29  3
Example 2.
12 
3
( x  2)  8
2
Example 3.
x 1  1
2 x  12
Assignment: p. 92-93, 33, 45 - 55
odds, 100 - 103.