Download Graph and analyze each function. Describe its

Graph and analyze each function. Describe its
domain, range, intercepts, end behavior,
continuity, and where the function is increasing
or decreasing.
Mid-Chapter Quiz: Lessons 2-1 through 2-3
Graph and analyze each function. Describe its
domain, range, intercepts, end behavior,
continuity, and where the function is increasing
or decreasing.
1. f (x) = 2x
1. f (x) = 2x3
SOLUTION: Evaluate the function for several x-values in its
domain.
3
x
−2
−1.5
−1
0
1
1.5
2
SOLUTION: Evaluate the function for several x-values in its
domain.
x
−2
−1.5
−1
0
1
1.5
2
f(x)
−16
−6.75
−2
0
2
6.75
16
f(x)
−16
−6.75
−2
0
2
6.75
16
Use these points to construct a graph.
Use these points to construct a graph.
The function is a monomial with an even degree and
a positive value for a.
All values of x are included in the graph, so the
function exists for all values of x, and D = (− ∞, ∞).
The function is a monomial with an even degree and
a positive value for a.
All values of y are included in the graph, so the
function exists for all values of y, and R = (− ∞, ∞).
All values of x are included in the graph, so the
function exists for all values of x, and D = (− ∞, ∞).
All values of y are included in the graph, so the
function exists for all values of y, and R = (− ∞, ∞).
The only time the graph intersects the axes is when
it goes through the origin, so the x- and y-intercepts
are both 0.
The only time the graph intersects the axes is when
it goes through the origin, so the x- and y-intercepts
are both 0.
The y-values approach negative infinity as x
approaches negative infinity, and positive infinity as x
approaches positive infinity, so
The y-values approach negative infinity as x
approaches negative infinity, and positive infinity as x
approaches positive infinity, so
and
and
.
There are no breaks, holes, or gaps in the graph, so it
is continuous for all real numbers.
.
As you read the graph from left to right, it is going up
from negative infinity to positive infinity, so the graph
is increasing on (− ∞, ∞).
There are no breaks, holes, or gaps in the graph, so it
is continuous for all real numbers.
As you read the graph from left to right, it is going up
from negative infinity to positive infinity, so the graph
is increasing on (− ∞, ∞).
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2. SOLUTION: As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on (− ∞, 0) and decreasing on (0, ∞).
As you read the graph from left to right, it is going up
from negative infinity to positive infinity, so the graph
Mid-Chapter
Quiz:
Lessons 2-1 through 2-3
is increasing on
(− ∞, ∞).
3. f (x) = 3x −8
2. SOLUTION: SOLUTION: Evaluate the function for several x-values in its
domain.
Evaluate the function for several x-values in its
domain.
x
−2
−1.5
−1
0
1
1.5
2
f(x)
−10.7
−3.4
−0.7
0
−0.7
−3.4
−10.7
x
−2
−1
−0.5
f(x)
0.01
3
768
0
0.5
1
2
768
3
0.01
Use these points to construct a graph.
Use these points to construct a graph.
Since the power is negative, the function will be
undefined at x = 0, and D = (− ∞, 0)∪(0, ∞).
The function is a monomial with an even degree and
a positive value for a.
The only values of y that are included in the graph
are greater than 0 through infinity, so R = (0, ∞).
All values of x are included in the graph, so the
function exists for all values of x, and D = (− ∞, ∞).
The graph never intersects either axis, so there are
no x- and y-intercepts.
The only values of y that are included in the graph
are negative infinity through 0, so R = (− ∞, 0].
The y-values approach zero as x approaches
negative or positive infinity, so
The only time the graph intersects the axes is when
it goes through the origin, so the x- and y-intercepts
are both 0.
and
.
The y-values approach negative infinity as x
approaches negative or positive infinity, so
and
.
The graph has an infinite discontinuity at x = 0.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going up again
from 0 to positive infinity, so the graph is increasing
on (− ∞, 0)∪(0, ∞).
There are no breaks, holes, or gaps in the graph, so it
is continuous for all real numbers.
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going down from
0 to positive infinity, so the graph is increasing on (− ∞, 0) and decreasing on (0, ∞).
3. f (x) = 3x −8
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SOLUTION: Evaluate the function for several x-values in its
4. SOLUTION: Evaluate the function for several x-values in its
domain.
x
−4
−2
f(x)
6.97
5.28
Page 2
As you read the graph from left to right, it is going up
from negative infinity to 0, and then going up again
from 0 to positive
infinity,
so the graph
increasing2-3
Mid-Chapter
Quiz:
Lessons
2-1 isthrough
on (− ∞, 0)∪(0, ∞).
As you read the graph from left to right, it is going
down from negative infinity to 0, and then going up
from 0 to positive infinity, so the graph is decreasing
on (− ∞, 0) and increasing on (0, ∞).
5. TREES The heights of several fir trees and the 4. areas under their branches are shown in the table.
SOLUTION: Evaluate the function for several x-values in its
domain.
x
−4
−2
−1
0
1
2
4
f(x)
6.97
5.28
4
0
4
5.28
6.97
a. Create a scatter plot of the data.
b. Determine a power function to model the data.
c. Predict the area under the branches of a fir tree
that is 7.6 meters high.
SOLUTION: Use these points to construct a graph.
a. Enter the data into a graphing calculator and
create a scatter plot.
The function is a power function with a rational
exponent and a positive value for a.
b. Use the power regression function on the
graphing calculator to find values for a and n.
All values of x are included in the graph, so the
function exists for all values of x, and D = (− ∞, ∞).
The only values of y that are included in the graph
are 0 through infinity, so R = [0, ∞).
2.3
f(x) = 1.37x
c. Graph the regression equation using a graphing
calculator. To predict the area under the branches of
a fir tree that is 7.6 meters high, use the value
function from the CALC menu on the graphing
calculator. Let x = 7.6.
The only time the graph intersects the axes is when
it goes through the origin, so the x- and y-intercepts
are both 0.
The y-values approach positive infinity as x
approaches negative or positive infinity, so
and
.
There are no breaks, holes, or gaps in the graph, so it
is continuous for all real numbers.
As you read the graph from left to right, it is going
down from negative infinity to 0, and then going up
from 0 to positive infinity, so the graph is decreasing
on (− ∞, 0) and increasing on (0, ∞).
The area under the branches of a fir tree that is 7.6
meters high is about 149.26 square meters.
5. TREES The heights of several fir trees and the eSolutions Manual - Powered by Cognero
areas under their branches are shown in the table.
Solve each equation.
6. = 13
Page 3
Mid-Chapter
Quiz:
Lessons
2-1tree
through
The area under
the branches
of a fir
that is 7.62-3
The solutions are 1 and 3.
meters high is about 149.26 square meters.
Solve each equation.
6. 8. = 13
+1=
SOLUTION: SOLUTION: Since the each side of the equation was raised to a
power, check the solution in the original equation.
x = 5 or x = −2. Since the each side of the equation
was raised to a power, check the solutions in the
original equation.
The solution is 32.4.
x = 5:
7. SOLUTION: x = −2:
x = 3 or x = 1. Since the each side of the equation
was raised to a power, check the solutions in the
original equation.
x =3
The solution is −2.
9. SOLUTION: x =1
The solutions are 1 and 3.
8. +1=
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Since the each side of the equation was raised to a
power, check the solution in the original equation.
Mid-Chapter Quiz: Lessons 2-1 through 2-3
The solution is −2.
So, the zeros are −2 and 13.
11. f (x) = 3x5 + 2x4 – x3
9. SOLUTION: SOLUTION: The degree of f (x) is 5, so it will have at most five
real zeros and four turning points.
Since the each side of the equation was raised to a
power, check the solution in the original equation.
The zeros are –1, 0, and
.
12. f (x) = x4 + 9x2 – 10
SOLUTION: The degree of f (x) is 4, so it will have at most four
real zeros and three turning points.
2
Let u = x .
The solution is 13.
State the number of possible real zeros and
turning points of each function. Then find all of
the real zeros by factoring.
10. f (x) = x2 – 11x – 26
SOLUTION: The degree of f (x) is 2, so it will have at most two
real zeros and one turning point.
Because ±
are not real zeros, f has two
distinct real zeros, −1 and 1.
13. MULTIPLE CHOICE Which of the following de
behavior of a polynomial of odd degree?
A
So, the zeros are −2 and 13.
5
4
11. f (x) = 3x + 2x – x
3
B
C
D
SOLUTION: eSolutions
Cognero
TheManual
degree- Powered
of f (x) by
is 5,
so it will
have at most five
real zeros and four turning points.
SOLUTION: Page 5
A polynomial of odd degree with either have an end
Because ± Quiz:
are not
real zeros,
two
Mid-Chapter
Lessons
2-1f has
through
distinct real zeros, −1 and 1.
2-3
13. MULTIPLE CHOICE Which of the following de
behavior of a polynomial of odd degree?
A
B
C
D
SOLUTION: A polynomial of odd degree with either have an end
a. Determine a model for the number of kilowatt
hours Crystal used as a function of the number of
months since January.
b. Use the model to predict how many kilowatt hours
Crystal will use the following January. Does this
answer make sense? Explain your reasoning.
SOLUTION: a. Sample answer: Since the data shows two
possible turning points, a third-degree polynomial
may be the best model to represent the data. Use the
cubic regression function on the graphing calculator.
Let x = 0 represent January.
The correct answer is D.
Describe the end behavior of the graph of each
polynomial function using limits. Explain your
reasoning using the leading term test.
14. f (x) = –7x4 – 3x3 – 8x2 + 23x + 7
SOLUTION: The degree is 4 and the leading coefficient is −7.
Because the degree is even and the leading
coefficient is negative,
3
2
f(x) = –2.707x + 41.392x – 141.452x + 238.176
b. Sample answer: Graph the regression equation
using a graphing calculator. To predict how many
kilowatt hours Crystal will use the following January,
use the CALC function. Since the following January
is 12 months from the first January, let x = 12.
15. f (x) = –5x5 + 4x4 + 12x2 – 8
SOLUTION: The degree is 5 and the leading coefficient is –5.
Because the degree is odd and the leading
coefficient is negative,
16. ENERGY Crystal’s electricity consumption
The amount of kilowatt hours Crystal will use the
following January is about −177.273. This answer
does not make sense because it is not possible to
consume negative kWh in a month.
measured in kilowatt hours (kWh) for the past 12
months is shown below.
a. Determine a model for the number of kilowatt
hours Crystal used as a function of the number of
months since January.
b. Use the model to predict how many kilowatt hours
Crystal will use the following January. Does this
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answer make sense? Explain your reasoning.
Page 6