1-2 Analyzing Graphs of Functions and Relations

6.

a. h(-1) b. h(1.5)

1-2 Analyzing Graphs of Functions and Relations c. h(2)

SOLUTION:

Use the graph of each function to estimate the indicated function values. Then confirm the estimate algebraically. Round to the nearest hundredth, if necessary.

The function value at x = ?1 appears to be about ?4. To confirm this estimate algebraically, find h(?1).

3. a. f (-8) b. f (-3) c. f (0)

SOLUTION: The function value at x = ?8 appears to be about 10. To confirm this estimate algebraically, findf (?8).

The function value at x = 1.5 appears to be about . Find h(1.5).

The function value at x = ?3 appears to be about 5. Findf (?3).

The function value at x = 0 appears to be about 2. Findf (0).

The function value at x = 2 appears to be about . Find h(2).

6. a. h(-1) b. h(1.5) c. h(2)

SOLUTION:

The function value at x = ?1 appears to be about ?4. eSolutTionoscMoannfuiraml - PthoiwseeresdtibmyaCtoeganlegroebraically, find h(?1).

Use the graph of h to find the domain and range of each function.

9. Page 1 SOLUTION: The arrows on the left and right sides of the graph

15. ENGINEERING Tests on the physical behavior of four metal specimens are performed at various

1-2 Analyzing Graphs of Functions and Relations temperatures in degrees Celsius. The impact energy,

or energy absorbed by the sample during the test, is measured in Joules. The test results are shown. Use the graph of h to find the domain and range of each function.

9. SOLUTION: The arrows on the left and right sides of the graph indicate that the graph will continue without bound in both directions. Therefore, the domain of h is (? , ).

The graph does not extend below h(0) or 2, but h(x) increases without bound for lesser and greater values of x. So, the range of h is [2, ).

12. SOLUTION: The closed dot at (7, ?1) indicates that x = 7 is in the domain of h. Although there is an open dot at (4, 1), the closed dot at (4, ?1) indicates that x = 4 is in the domain of h. The arrow to the left indicates that the graph will continue without bound. Therefore, the domain of h is (? , 7].

The closed dots at (4, ?1) and (7, ?1) indicate that y = ?1 is in the range of h. The closed dot at (5, 1) indicates that y = 1 is not in the range of h. The arrow to the left indicates that the graph will continue without bound. Therefore, the range of h is [?1] (1, ).

15. ENGINEERING Tests on the physical behavior of four metal specimens are performed at various temperatures in degrees Celsius. The impact energy, or energy absorbed by the sample during the test, is measured in Joules. The test results are shown.

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a. State the domain and range of each function. b. Use the graph to estimate the impact energy of each metal at 0?C.

SOLUTION: a. The arrows on the left and right sides of the graph that corresponds to the copper specimen indicate that the graph will continue without bound. So, the domain is all real numbers, or [?, ]. The impact energy of the specimen for all temperatures from ? 150?C to 150?C appears to be about 1.75 joules, so the range is [1.75]

The arrows on the left and right sides of the graph that corresponds to the aluminum specimen indicate that the graph will continue without bound. So, the domain is [?, ]. The graph does not extend above f(?100) = 1.5 or below f (125) = 0.6. So, the range is [0.6, 1.5].

The arrows on the left and right sides of the graph that corresponds to the zinc specimen indicate that the graph will continue without bound. So, the domain is [?, ]. The graph does not extend above f(?100) = 0.5 or below f (100) = 1.3. So, the range is [0.5, 1.3].

The arrows on the left and right sides of the graph that corresponds to the steel specimen indicate that the graph will continue without bound. So, the domain is [-, ]. The graph does not extend above f(?125) = 0.2 or below f (100) = 1.75. So, the range is [0.2, 1.75].

Note that absolute zero, the coldest temperature possible, is about ?273.15?C. Therefore, in the context of this problem, the true domain is [?273.15, ].

b. Estimate the function value at x = 0 for each curve.

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possible, is about ?273.15?C. Therefore, in the context of this problem, the true domain is [?273.15, ].

1-2 bA. nEastlimyzatientgheGfurnactpiohnsvaolufeFaut xn=ct0iofonrseaacnhd Relations Therefore, the zero of f is 0.

curve.

When x = 0, the copper, aluminum, zinc, and steel specimens appear to have impact energies of about 1.75 J, 1.2 J, 0.5 J, and 1.5 J, respectively. Use the graph of each function to find its yintercept and zero(s). Then find these values algebraically.

18. SOLUTION: From the graph, it appears that f (x) intersects the yaxis at approximately (0, 0), so the y -intercept is 0. Findf (0).

Therefore, the y-intercept is 0. From the graph, it appears that there is an xintercept at 0. Let f (x) = 0 and solve for x.

21. SOLUTION: From the graph, it appears that f (x) intersects the yaxis at approximately (0, ?2), so the y -intercept is ? 2. Findf (0).

Therefore, the y-intercept is ?2. From the graph, the x-intercepts appear to be at about ? and . Let f (x) = 0 and solve for x.

Therefore, the zeros of f are and . Use the graph of each equation to test for symmetry with respect to the x-axis, y-axis, and the origin. Support the answer numerically. Then confirm algebraically.

Therefore, the zero of f is 0.

21. SOLUTION:

eSolutFiornosmMatnhuealg-rPaopwhe,reitdabpypCeoagrnsertohat f (x) intersects the y axis at approximately (0, ?2), so the y -intercept is ? 2. Findf (0).

24.

SOLUTION: The graph appears to be symmetric with respect to the x-axis, y -axis, and origin because there appears to be mirror images about the axes and the origin. Also, for every point (x, y) on the graph, there is a point (x, ?y), a point (?x, y), and a point (?x, ?y), respectively. Make a table of values to support each part of this conjecture.

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to be mirror images about the axes and the origin.

Also, for every point (x, y) on the graph, there is a

Confirm algebraically.

1-2

point (x, ?y), a point (?x, y), and a point (?x, ?y),

rAesnpaecltyivzeilny.gMGakreaaptahbsleooff Fvaulunecs ttoiosnupspaornt deacRhelations

part of this conjecture.

(?x)2 is equidistant to x2, so equivalent to x2 + 4y2 = 16.

(-x)2

+

4y 2

=

16

is

The positive y -values produce the same x-values as their corresponding ?y-values, so x2 + 4(-y)2 = 16 is equivalent to x2 + 4y2 = 16 and the graph is symmetric with respect to the x-axis.

Confirm algebraically. (?y )2 is equidistant to y 2, so x2 + 4(-y)2 = 16 is equivalent to x2 + 4y2 = 16.

Because (-x) 2 + 4(-y)2 = 16 is equivalent to x 2 + 4y2 = 16, the graph is symmetric with respect to the origin.

27.

SOLUTION: The graph appears to be symmetric with respect to the x-axis, y -axis, and origin because there appears to be mirror images about all three. Also, for every point (x, y) on the graph, there is a point (x, ?y), a point (?x, y), and a point (?x, ?y), respectively. Make a table of values for each part of this conjecture.

The positive x-values produce the same y -values as their corresponding ?x-values, so (-x)2 + 4y2 = 16 is equivalent to x2 + 4y2 = 16 and the graph is symmetric with respect to the y -axis.

Confirm algebraically. (?x)2 is equidistant to x2, so (-x)2 + 4y2 = 16 is equivalent to x2 + 4y2 = 16.

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The positive y-values produce the same x-values as their corresponding ?y-values, so 9x2 ? 25(-y)2 P=ag1e 4 is equivalent to 9x2 ? 25y2 = 1 and the graph is symmetric with respect to the x-axis.

1-2 Analyzing Graphs of Functions and Relations

The positive y-values produce the same x-values as their corresponding ?y-values, so 9x2 ? 25(-y)2 = 1 is equivalent to 9x2 ? 25y2 = 1 and the graph is symmetric with respect to the x-axis.

Because 9(-x) 2 - 25(-y)2 = 1 is equivalent to 9x2 ? 25y2 =1, the graph is symmetric with respect to the origin.

Confirm algebraically. (?y )2 is equivalent toy 2, so 9x2 ? 25(-y)2 = 1 is equivalent to 9x2 ? 25y2 = 1.

The positive x-values produce the same y -values as their corresponding ?x-values, so 9(-x)2 ? 25y2 = 1 is equivalent to 9x2 ? 25y2 =1 and the graph is symmetric with respect to the y -axis.

Confirm algebraically. (?x)2 is equivalent to x2, so 9(-x)2 ? 25y 2 = 1 is equivalent to 9x2 ? 25y2 =1.

30. SOLUTION: The graph does not appear to be symmetric with respect to the x-axis, y -axis, or origin because there are no mirror images with respect to these areas of the graph.

Because neither ?y = x3 ? 2x2 + 3x ? 4, y = ?x3 ? 2x2 ? 3x ? 4, nor ?y = ?x3 ? 2x2 ? 3x ? 4 are equivalent to y = x3 ? 2x2 + 3x ? 4, the graph is not symmetric with respect to the x-axis, y -axis, or origin, respectively.

33. SOLUTION: The graph appears to be symmetric with respect to the y-axis because there appears to be a mirror image about the y-axis. Also, for every point (x, y) on the graph, there is a point (?x, y). Make a table of values to support this conjecture.

Because 9(-x) 2 - 25(-y)2 = 1 is equivalent to 9x2 ? eSolut2io5nys2M=an1u,atlh-ePogwraeprehd ibsy sCyomgnmeroetric with respect to the

origin.

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