Transcript 2-1 Power & Radical Functions Extra Notes and

LESSON 2–1
Power and Radical
Functions
Over Chapter 1
Describe the end behavior of f(x) = 4x 4 + 2x – 8.
A.
B.
C.
D.
C > 1 expands it
Over Chapter 1
0 < C < 1 compresses it
Identify the parent function f(x) of g(x) = 2|x – 3| + 1.
Describe how the graphs of g(x) and f(x) are
related.
A. f(x) = | x |; f(x) is translated 3 units right, 1 unit
up and expanded vertically to graph g(x).
B. f(x) = | x |; f(x) is translated 3 units right, 1 unit
up and expanded horizontally to graph g(x).
C. f(x) = | x |; f(x) is translated 3 units left, 1 unit
up and expanded vertically to graph g(x).
D. f(x) = | x |; f(x) is translated 3 units left, 1 unit
down and expanded horizontally to graph g(x).
Over Chapter 1
Find [f ○ g](x) and [g ○ f ](x) for f(x) = 2x – 4 and
g(x) = x 2.
A. (2x – 4)x 2; x 2(2x – 4)
B. 4x 2 – 16x + 16; 2x 2 – 4
C. 2x 2 – 4; 4x 2 – 16x + 16
D. 4x 2 – 4; 4x 2 + 16
Over Chapter 1
Evaluate f(2x) if f(x) = x 2 + 5x + 7.
A. 2x 2 + 10x + 7
B. 2x 3 + 10x 2 + 7
C. 4x 2 + 10x + 7
D. 4x 2 + 7x + 7
Analyze Monomial Functions
A. Graph and analyze
. Describe the
domain, range, intercepts, end behavior, continuity,
and where the function is increasing or decreasing.
Evaluate the function for several x-values in its
domain. Then use a smooth curve to connect each of
these points to complete the graph.
Analyze Monomial Functions
D = (–∞, ∞); R = [0, ∞);
intercept: 0;
end behavior:
continuity: continuous for all real numbers;
decreasing: (–∞, 0); increasing: (0, ∞)
Analyze Monomial Functions
Answer: D = (–∞, ∞); R = [0, ∞); intercept: 0;
continuous for all real numbers;
decreasing: (–∞, 0) , increasing: (0, ∞)
Analyze Monomial Functions
B. Graph and analyze f(x) = –x 5. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing or
decreasing.
Analyze Monomial Functions
D = (–∞, ∞); R = (–∞, ∞);
intercept: 0;
end behavior:
continuity: continuous for all real numbers;
decreasing: (–∞, ∞)
Analyze Monomial Functions
Answer: D = (–∞, ∞); R = (–∞, ∞); intercept: 0;
continuous for all real numbers;
decreasing: (–∞, ∞)
Describe where the graph of the function
f(x) = 2x 4 is increasing or decreasing.
A. increasing: (–∞, ∞)
B. decreasing: (–∞, 0) , increasing: (0, ∞)
C. decreasing: (–∞, ∞)
D. increasing: (–∞, 0), decreasing: (0, ∞)
Functions with Negative Exponents
A. Graph and analyze f(x) = 2x –4. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing or
decreasing.
Functions with Negative Exponents
intercept: none;
end behavior:
continuity: infinite discontinuity at x = 0;
increasing: (–∞, 0); decreasing: (0, ∞)
Functions with Negative Exponents
Answer: D = (– ∞, 0)  (0, ∞); R = (0, ∞); no intercept
;
infinite discontinuity at x = 0;
increasing: (–∞, 0), decreasing: (0, ∞);
Functions with Negative Exponents
B. Graph and analyze f(x) = 2x –3. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing or
decreasing.
Functions with Negative Exponents
D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞);
intercept: none;
end behavior:
continuity: infinite discontinuity at x = 0;
decreasing: (–∞, 0) and (0, ∞)
Functions with Negative Exponents
Answer: D = (–∞, 0)  (0, ∞); R = (–∞, 0)  (0, ∞);
no intercept
;
infinite discontinuity at x = 0;
decreasing: (–∞, 0) and (0, ∞)
Describe the end behavior of the graph of f(x) = 3x –5.
A.
B.
C.
D.
Rational Exponents
A. Graph and analyze
. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing or
decreasing.
Rational Exponents
D = [0, ∞); R = [0, ∞);
intercept: 0;
end behavior:
continuity: continuous on [0, ∞);
increasing: [0, ∞)
Rational Exponents
Answer: D = [0, ∞); R = [0, ∞); intercept: 0;
;
continuous on [0, ∞);
increasing: [0, ∞)
Rational Exponents
B. Graph and analyze
. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing or
decreasing.
Rational Exponents
D = (0, ∞); R = (0, ∞);
intercept: none;
end behavior:
continuity: continuous on (0, ∞);
decreasing: (0, ∞)
Rational Exponents
Answer: D = (0, ∞); R = (0, ∞); no intercept
;
continuous on (0, ∞);
decreasing: (0, ∞)
Describe the continuity of the function
A. continuous for all real numbers
B. continuous on
C. continuous on (0, ∞]
D. continuous on [0, ∞)
and
.
Power Regression
A. ANIMALS The following data represent the body
length L in centimeters and the mass M in kilograms
of several African Golden cats being studied by a
scientist. Create a scatter plot of the data.
Power Regression
Answer:
The scatter plot appears to resemble the square root
function which is a power function.
Power Regression
B. ANIMALS The following data represent the
body length L in centimeters and the mass M in
kilograms of several African Golden cats being
studied by a scientist. Determine a power function
to model the data. Describe its end behavior
Power Regression
Using the PwrReg tool on a graphing calculator yields
y = 0.018x1.538. The correlation coefficient r for the
data, 0.875, suggests that a power regression
accurately reflects the data.
Answer: L = 0.018M 1.538
Power Regression
C. ANIMALS The following data represent the
body length L in centimeters and the mass M in
kilograms of several African Golden cats being
studied by a scientist. Use the data to predict the
mass of an African Golden cat with a length of
77 centimeters.
Power Regression
Use the CALC feature on the calculator to find f(77).
The value of f(77) is about 14.1, so the mass of an
African Golden cat with a length of 77 centimeters is
about 14.1 kilograms.
Answer: 14.1 kg
AIR The table shows the amount of air f(r) in cubic
inches needed to fill a ball with a radius of r
inches. Determine a power function to model the
data.
A. f(r) = 5.9r 2.6
B. f(r) = 0.6r 0.3
C. f(r) = 19.8(1.8)r
D. f(r) = 5.2r 2.9
Graph Radical Functions
A. Graph and analyze
. Describe the
domain, range, intercepts, end behavior,
continuity, and where the function is increasing or
decreasing.
Graph Radical Functions
D = [0, ∞); R = [0, ∞);
intercept: 0;
continuous on [0, ∞);
increasing: [0, ∞)
Graph Radical Functions
Answer: D = [0, ∞); R = [0, ∞); intercept: 0;
;
continuous on [0, ∞);
increasing: [0, ∞)
Graph Radical Functions
B. Graph and analyze
. Describe the
domain, range, intercepts, end behavior, continuity,
and where the function is increasing or decreasing.
Graph Radical Functions
D = (–∞, ∞); R = (–∞, ∞);
x-intercept:
, y-intercept: about –0.6598;
;
continuous for all real numbers;
increasing: (–∞, ∞)
Graph Radical Functions
Answer: D = (–∞, ∞) ; R = (–∞, ∞) ; x-intercept:
y-intercept: about –0.6598;
;
continuous for all real numbers;
increasing: (–∞, ∞)
,
Find the intercepts of the graph of
A. x-intercept:
B. x-intercepts:
C. x-intercept:
D. x-intercepts:
, y-intercept:
, y-intercept:
, y-intercept:
, y-intercept –4
.
Solve Radical Equations
A. Solve
.
original equation
Isolate the radical.
Square each side to
eliminate the radical.
Subtract 28x and 29
from each side.
Factor.
Factor.
x – 5 = 0 or x + 1 = 0
x =5
x = –1
Zero Product Property
Solve.
Solve Radical Equations
Answer: –1, 5
Check
x = –1
x=5
10 = 10 
–2 = –2 
A check of the solutions in the original equation
confirms that the solutions are valid.
Solve Radical Equations
B. Solve
.
original equation
Subtract 8 from each side.
Raise each side to the third
power. (The index is 3.)
Take the square root of each side.
x = 10 or –6
Add 2 to each side.
A check of the solutions in the original equation
confirms that the solutions are valid.
Answer: 10, –6
Solve Radical Equations
C. Solve
.
original equation
Square each side.
Isolate the radical.
Square each side.
(x – 8)(x – 24) = 0
x – 8 = 0 or x – 24 = 0
Distributive Property
Combine like terms.
Factor.
Zero Product Property
Solve Radical Equations
x= 8
x = 24
Solve.
One solution checks and the other solution does not.
Therefore, the solution is 8.
Answer: 8
Solve
A. 0, 5
B. 11, –11
C. 11
D. 0, 11
.
LESSON 2–1
Power and Radical
Functions