Query 23

#$&*

course Mth 163

4-9-13 1:00

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

023. `query 23

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Question: `qQuery problem 2.

Describe the sum of the two graphs.

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Your solution:

ok

confidence rating #$&*: 3

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Given Solution:

`a** The 'black' graph takes values 8, 3, 0, -1, 0, 3, 8 at x = -3, -2, -1, 0, 1, 2, 3.

The 'blue' graph takes approximate values 1.7, .8, .2, -.1, -.4, -.6, -.8 at the same x values.

The 'blue' graph takes value zero at approximately x = -.4.

The sum of the two graphs will coincide with the 'blue' graph where the 'black' graph is zero, which occurs at x = -1 and x

= 1.

The sum will coincide with the 'black' graph where the 'blue' graph is zero, which occurs at about x = -.4. **

STUDENT QUESTION:

I don't understand where you are getting these numbers.

INSTRUCTOR RESPONSE: The graph in the stated problem appears below. The x and y scales are marked in units. You should begin by orienting yourself to these graphs: convince yourself that the points marked on the graphs have the coordinates quoted in the given solution.

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Self-critique (if necessary):

ok

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Self-critique rating: ok

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Question: `qWhere it is the sum graph higher than the 'black' graph, and where is it lower? Answer by giving specific intervals.

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Your solution:

The sum is higher than the black graph when the blue is positive (-3, -4) and lower where the blue graph is negative

confidence rating #$&*: 3

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Given Solution:

`a** The sum of the graphs is higher than the 'black' graph where the 'blue' graph is positive, lower where the 'blue' graph

is negative.

The 'blue' graph is positive on the interval from x = -3 to x = -.4, approx.. This interval can be written [-3, -.4), or -3 <=

x < -.4. **

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Self-critique (if necessary):

ok

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Self-critique rating: ok

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Question: `qWhere it is the sum graph higher than the 'blue' graph, and where is it lower? Answer by giving

specific intervals.

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Your solution:

ok

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`a** The sum of the graphs is higher than the 'blue' graph where the 'black' graph is positive, lower where the 'black'

graph is negative.

The 'black' graph is positive on the interval from x = -1 to x = 1, not including the endpoints of the interval. This interval

can be written (-1, 1) or -1 < x < 1. **

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Self-critique (if necessary):

ok

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Self-critique rating: ok

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Question: `qWhere does thus sum graph coincide with the 'black' graph, and why? Give your estimate of the

specific coordinates of the point or points where this occurs.

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Your solution:

ok

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`a** The sum coincides with the 'black' graph where the 'blue' graph is zero, which occurs at about x = -.4. The

coordinates would be about (-.4, -.7), on the 'black' graph. **

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Self-critique (if necessary):

ok

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Self-critique rating: ok

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Question: `qWhere does thus sum graph coincide with the 'blue' graph, and why? Give your estimate of the

specific coordinates of the point or points where this occurs.

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Your solution:

Ok

confidence rating #$&*: 3

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Given Solution:

`a** The sum coincides with the 'blue' graph where the 'black' graph is zero, which occurs at x = -1 and x = 1. The

coordinates would be about (-1, .2) and (1, -.4), on the 'blue' graph. **

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Self-critique (if necessary):

ok

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Self-critique rating: ok

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Question: `qQuery problem 3

Describe the quotient graph obtained by dividing the 'black' graph by the 'blue' graph. You should answer the following

questions:

Where it is the quotient graph further from the x axis than the 'black' graph, and where is it closer? Answer by giving

specific intervals, and explaining why you believe these to be the correct intervals.

Where it is the quotient graph on the same side of the x axis as the 'black' graph, and where is it on the opposite side,

and why? Answer by giving specific intervals.

Where does thus quotient graph coincide with the 'black' graph, and why? Give your estimate of the specific coordinates

of the point or points where this occurs.

Where does the quotient graph have vertical asymptote(s), and why? Describe the graph at each vertical asymptote.

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Your solution:

ok

confidence rating #$&*: 3

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Given Solution:

`a** The 'black' graph is periodic, passing through 0 at approximately x = -3.1, 0, 3.1, 6.3. This graph has peaks with y

= 1.5, approx., at x = 1.6 and 7.8, approx., and valleys with y = -1.5 at x = -1.6 and x = 4.7 approx.

The 'blue' graph appears to be parabolic, passing thru the y axis at x = -1 and reaching a minimum value around y = -1.1

somewhere near x = 1. This graph passes thru the x axis at x = 5.5, approx., and first exceeds y = 1 around x = 7.5.

The quotient will be further from the x axis than the 'black' graph wherever the 'blue' graph is within 1 unit of the origin,

since division by a number whose magnitude is less than 1 gives a result whose magnitude is greater than the number being

divided. This will occur to the left of x = 1, and between about x = 2 and x = 7.5.

Between about x = 0 and x = 1 the 'blue' graph is more than 1 unit from the x axis and the quotient graph will be closer to

the x axis than the 'black' graph. The same is true for x > 7.5, approx..

The 'black' graph is zero at or near x = -3.1, 0, 3.1, 6.3. At both of these points the 'blue' graph is nonzero so the

quotient will be zero.

The 'blue' graph is negative for x < 5.5, approx.. Since division by a negative number gives us the opposite sign as the

number being divided, on this interval the quotient graph will be on the opposite side of the x axis from the 'black' graph.

The 'blue' graph is positive for x > 5.5, approx.. Since division by a positive number gives us the same sign as the number

being divided, on this interval the quotient graph will be on the same side of the x axis as the 'black' graph.

The quotient graph will therefore start at the left with positive y values, about 3 times as far from the x axis as the 'black'

graph (this since the value of the 'blue' graph is about -1/3, and division by -1/3 reverses the sign and gives us a result with

3 times the magnitude of the divisor).

The quotient graph will have y value about 2.5 at x = -1.6, where the 'black' graph 'peaks', but the quotient graph will

'peak' slightly to the left of this point due to the increasing magnitude of the 'blue' graph.

The quotient graph will then reach y = 0 / (-1) = 0 at x = 0 and, since the 'black' graph then becomes positive while the

'blue' graph remains negative, the quotient graph will become negative.

Between x = 0 and x = 2 the magnitude of the 'blue' graph is a little greater than 1, so the quotient graph will be a little

closer to the x axis than the 'black' graph (while remaining on the other side of the x axis).

At x = 3.1 approx. the 'black graph is again zero, so the quotient graph will meet the x axis at this point.

Past x = 3.1 the quotient graph will become positive, since the signs of both graphs are negative. As we approach x =

5.5, where the value of the 'blue' graph is zero, the quotient will increase more and more rapidly in magnitude (this since

the result of dividing a negative number by a negative number near zero is a large positive number, larger the closer the

divisor is to zero). The result will be a vertical asymptote at x = 5.5, with the y value approaching +infinity as x

approaches 5.5 from the left.

Just past x = 5.5 the 'blue' values become positive. Dividing a negative number by a positive number near zero results in a

very large negative value, so that on this side of x = 5.5 the asymptote will rise up from -infinity.

The quotient graph passes through the x axis near x = 6.3, where the 'black' graph is again zero. To the right of this point

both graphs have positive values and the quotient graph will be positive.

Around x = 7.5, where the 'blue' value is 1, the graph will coincide with the 'black' graph, giving us a point near (7.5, 1.3).

Past this point the 'blue' value is greater than 1 so that the quotient graph will become nearer the x axis than the 'black'

graph, increasingly so as x (and hence the 'blue' value) increases. This will result in a 'peak' of the quotient graph

somewhere around x = 7.5, a bit to the left of the peak of the 'black' graph. **

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Self-critique (if necessary):

ok

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Self-critique rating:

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Question: `qQuery problems 7-8

Sketch the graph of y = x^2 - 2 x^4 by first sketching the graphs of y = x^2 and y = -2 x^4.

How does the result compare to the graph of y = x^2 - x^4, and how do you explain the difference?

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Your solution:

x X^2 -2x^4 X^2-2x^4 X^2-x^4

1 1 -2 -1 0

2 4 -32 -28 -12

3 9 -162 -153 -72

4 16 -512 -496 -240

5 25 -1250 -1225 -600

… ok…

confidence rating #$&*: 3

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Given Solution:

`a** At x = 0, 1/2, 1 and 2 we have x^2 values 0, 1/4, 1 and 4, while -x^4 takes values 0, -1/16, -1 and -16, and -2

x^4 takes values 0, -1/8, -2 and -32.

All graphs clearly pass through the origin.

The graphs of y = x^2 - x^4 and y = x^4 - 2 x^4 are both increasingly negative at far right and far left.

Graphical addition will show that y = x^2 - x^4 takes value 0 and hence passes thru the x axis when the graphs have

equal but opposite y values, which occurs at x = 1 and x = -1. To the left of x = -1 and to the right of x = 1 the negative

values of -x^4 overwhelm the positive values of x^2 and the sum graph will be increasingly negative, with values

dominated by -x^4. Near x = 0 the graph of y = -x^4 is 'flatter' than that of y = x^2 and the x^2 values win out, making

the sum graph positive.

y = x^2 - 2 x^4 will take value 0 where the graphs are equal and opposite in value; this occurs somewhere between x =

.8 and x = .9, and also between x = -.9 and x = -.8, which places the zeros closer to the y axis than those of the graph of

y = x^2 - x^4. The graph of y = -2 x^4 is still flatter near x = 0 than the graph of y = x^2, but not as flat as the graph of

y = -x^4, so while the sum graph will be positive between the zeros the values won't be as great. Outside the zeros the

sum graph will be increasingly negative, with values dominated by -2x^4. **

The graphs you constructed, based on the basic points and behaviors of the functions y = x^2, y = -x^4 and y = -2 x^4, should have had the same characteristics and basic properties of those in the series of figures below:

The figure below depicts the graphs of y = x^2 and y = -x^4, with x and y gridlines at unit intervals.

The graph of y = x^2 - x^4 is superimposed below:

The figure below depicts the graphs of y = x^2, y = -2 x^4 and y = x^2 - 2 x^4:

The graphs of y = x^2 - x^4 and y = x^2 - 2 x^4 are depicted below:

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `qHow does the shape of the graph change when you add x to get y = -2 x^4 + x^2 + x, and how

do you explain this change?

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Your solution:

ok

confidence rating #$&*: 3

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Given Solution:

`a** At x = 0 there is no change in the y value, so the graph still passes through (0, 0).

As x increases through positive numbers we will have to increase the y values of y = x^2 -2 x^4 by greater and greater

amounts. So it will take a little longer for the negative values of -2 x^4 to 'overwhelm' the positive values of x^2 + x than

to overcome the positive values of x^2 and the x intercept will shift a bit to the right.

As we move away from x = 0 through negative values of x we will find that the positive effect of y = x^2 is immediately

overcome by the negative values of y = x, so there is no x intercept to the left of x = 0.

The graph in fact stays fairly close to the graph of y = x near (0, 0), gradually moving away from that graph as the values

of x^2 and -2 x^4 become more and more significant. **

The graphs you constructed, based on the basic points and behaviors of the functions y = x^2, y = -x^4 and y = -2 x^4, should have had the same characteristics and basic properties of those in the series of figures below:

The figure below depicts the graphs of y = x^2 and y = -x^4, with x and y gridlines at unit intervals.

The graph of y = x^2 - x^4 is superimposed below:

The figure below depicts the graphs of y = x^2, y = -2 x^4 and y = x^2 - 2 x^4:

The graphs of y = x^2 - x^4 and y = x^2 - 2 x^4 are depicted below:

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Self-critique (if necessary):

ok

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Self-critique rating: ok

If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.

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Question: The graphs of two linear functions are depicted below. The first function f(x) has x-intercept (-8/3, 0) and y-intercept (0, 4), while the second function g(x) has y-intercept (0, 1) and slope -1/2.

You could easily find the formulas for these two functions and answer subsequent questions using these formulas, but don't do that. The task here, as you have seen, is to be able to construct graphs of combinations of functions. The purpose is to give you a deeper understanding of such functions and enhance your ability to think about and visualize complex trends.

Construct an approximate graph of the sum of these two functions.

At what approximate x coordinate do you estimate your graph of the sum intercepts the x axis?

Construct an approximate graph of the difference function f(x) - g(x).

At what approximate x coordinate do you estimate your graph of the difference function intercepts the x axis?

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Your solution:

x Y=1.5x+4 Y=-.5x+1 1.5x+4-.5x+1

0 4 1 5

1 5.5 .5 6

2 7 0 7

3 8.5 -.5 8

4 10 -1 9

5 11.5 -1.5 10

The function intercepts the graph at (-5,0).

X (1.5x+4)-(-.5x+1)

-1.5 0

0 3

1 5

2 7

3 9

4 11

5 13

The function intercepts the graph at (-1.5, 0)

@&

You appear to be relying on formulas for the functions and on numerical calculations rather than constructions.

You should be able to answer these questions without relying on formulas or tables of values. If you use a table of values you are calculating, rather than constructing, the graph.

On this question you should mark the scale of each axis, then plot the two points for each linear function. A construction will not estimate any other y values.

A construction would not rely on the equations of the functions either.

The point of this is that in order to understand and visualize trends in data, there will be cases where you need to look at two graphs and quickly visualize the graph of their sum, different, quotient, product or composite.

To determine where the sum graph is zero, you need to understand that this will occur when the graph of the sum intercepts the x axis, which happens when the y values are equal and opposite. In constructing the graph, you would be expected to have visually observed the graphs of the two functions to see where this is so, without looking at any x or y values. Once you locate the point at which you think this occurs, you would then mark the corresponding point of the x axis as the location of the zero. To determine the zeros of the sum function, here would be no need to estimate the y values of the two functions, because all we are interested in is the x value of the zero.

To construct the graph you would want to see the y values for a variety of x values without estimating them. If you see the two y values it is possible to then visualize what the sum of these values would look like, and plot the corresponding point. Similarly when constructing the difference graph you can visualize the two values and their difference.

A more subtle process of visualization would allow you to estimate where the sum or difference graph reaches a relative maximum or minimum (a point that's higher or lower than the other points in the immediate neighborhood). To do this you would need to think about how the relative slopes of the two functions affect the sum or difference graph.

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confidence rating #$&*: 3

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Question: For the graphs of the preceding, sketch the graph of the product function f(x) * g(x).

On what interval(s) is the graph of the product function positive, and on what intervals negative?

How could you have determined these intervals, based on the given graphs, without actually having sketched the graph of the product function?

On what interval(s) is the graph of the product function increasing, and on what intervals decreasing?

For large negative x, is the product function positive or negative, and how could you tell by just looking at the graphs of f(x) and g(x)?

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Your solution:

X -.75x^2-.5x+4

1 2.75

2 0

3 -4.25

4 -10

5 -17.25

The graph is positive from x=-2.66 to 2.

The graph of (1.5x+4) is positive and the graph of (-.5x+1) is negative.

The graph is increasing from negative infinity to 0 and decreasing from 0 to infinity.

The large negative x behavior of the graph is negative.

@&

Again you have relied on formulas and tables rather than construction. Your answers should follow from visual examination of the two original graphs. It would be legitimate to calculate, based on geometric reasoning with similar triangles, where each of the linear functions is zero, though it would be perfectly OK to just use good estimates based on your graphs.

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confidence rating #$&*: 2

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Question: For the same functions f(x) and g(x), sketch the graph of the quotient function f(x) / g(x).

Describe all vertical and horizontal asymptotes of your graph.

On what intervals is your graph positive, and on what intervals negative?

How could you have answered this question by just looking at the graphs of the two functions?

Given just the graphs of the two functions, how would you determine whether and where the given quotient function has one or more vertical asymptotes?

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Your solution:

There is a vertical Asymptote at x=2.

There is a horizontal asymptote at y=-3.

The graph is positive from -2.66 to infinity and negative from -infinity to -2.66 and 2 to infinity(never passing through -3)

1.5x+4 is positive and -.5x+1 is negative.

@&

It isn't clear how these results were obtained. In particular you should indicate how your answers are related to the graphs of the two linear functions, with no reference to the formulas for those functions or to results obtained from those formulas..

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confidence rating #$&*: 1

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Question: In the graph below the linear function is f(x), the nonlinear function g(x).

Think about the graphs of the functions f(x) + g(x), f(x) * g(x) and f(x) / g(x).

What will be the x intercepts of each of these functions?

On what intervals will each function be positive?

What will be the sign of each of these functions for large positive x, and for large negative x?

What will be the horizontal and vertical asymptotes of the quotient function?

Describe the behavior of the quotient function near each of its vertical asymptotes.

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Your solution:

The equation of the linear equation is y=1x+3. The equation of g(x) is (this is as close as I could get it. I wasn’t sure as to how to make the graph more accurate) y= -(x+3)(x-2)^2.

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This is good, and it's by no means trivial to arrive at these results, but they aren't related to the methods being used in this section, which concentrate on constructions as opposed to estimated values and formulas.

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F(x)+g(x)

The entire graph will move down 3 units resulting in only one real zero

The graph will be positive from negative infinity to negative 3

The end behavior of negative infinity is positive and the end behavior of infinity is negative

F(x)*g(x)

The graph will have 4 roots (with the equations I made for each graph the roots would be at -3, a double at 2, and 3)

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You don't need equations to determine where these roots are or whether single, double, etc..

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The graph is positive at negative infinity to 3.

The end behavior of negative infinity is negative and positive infinity negative

F(x)/g(x)

The graph will have no real roots

It will not have positive parts

The horizontal asymptote (with the equations I made) is at y=0

The vertical asymptote is at x=2

The end behavior as x approaches negative infinity is 0 and the end behavior as x approaches infinity is 0.

@&

These characteristics have to be related to characteristics of the graphs of the original functions, and to answer you would need to at least briefly describe this relationship.

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confidence rating #$&*: 0

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Self-critique rating: I had trouble with this problem

@&

The goal here is to be able to construct the graph of a sum, difference, quotient or product function from the graphs of two functions, whatever the shapes of the graphs. This is an important and powerful ability, as you will discover again and again in your subsequent mathematics and science education and in any career that relies on it. It is an ability which will distinguish you from most or your peers, who will be limited without it. Unlike most students, you have the aptitude and background to develop this ability, and I urge you to do so.

Overall you are bypassing these skills by relying on formulas (which you are using very well) and tables (which should not be used extensively in this course, and not at all in these problems) to answer questions that should be answered by visual examination, construction and analysis. (See also the notes I posted in the gradebook related to your last test, where on a number of problems you made inappropriate use of tables as opposed to the basic-points picture).

I recommend that you submit a revision of your solutions to these problems. In order to prepare to revise them, I suggest that you answer the following questions, and include your answers in the revised document:

In general, whatever functions f(x) and g(x) appear on a graph, and whether those functions can be described by formulas or not:

How can you determine the zeros of f(x) + g(x) from the graphs of f(x) and g(x)?

How can you determine where f(x) + g(x) is positive and where it is negative?

How can you determine the far-right and far-left behavior of f(x) + g(x)?

How can you estimate where f(x) + g(x) has a relative maximum or minimum? (this is a more challenging question, but if you can understand this the rewards are very benficial and can resonate throughout your subsequent education).

How can you use these characteristics to construct a reaonable graph of f(x) + g(x)?

How can you determine the zeros of f(x) * g(x) from the graphs of f(x) and g(x)?

How can you determine where f(x) * g(x) is positive and where it is negative?

How can you determine the far-right and far-left behavior of f(x) * g(x)?

How can you estimate where f(x) * g(x) has a relative maximum or minimum? (this is even more challenging than the same question for the sum graph, but it's worth at least some thought, if for no other reason than to get the question in your head).

How can you use these characteristics to construct a reaonable graph of f(x) * g(x)?

How can you determine the zeros of f(x) / g(x) from the graphs of f(x) and g(x)?

How can you determine the horizontal and vertical asymptotes of f(x) / g(x) from the graphs of f(x) and g(x)?

How can you determine where f(x) / g(x) is positive and where it is negative?

How can you determine the far-right and far-left behavior of f(x) / g(x)?

How can you use these characteristics to construct a reaonable graph of f(x) / g(x)?

&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).

Be sure to include the entire document, including my notes.

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