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- Reading Assignment: pp. 325-top 329
- Reading Assignment: pp. 210-211 (Examples 1-2), p. 213 (Example 4), p. 219 (Examples 2-5)
- Reading Assignment: bottom of p. 177 and top of p. 178 (proof of Sum/Difference Rule), pp. 186-187 (expls. 1-3), p. 188 (expls. 4-5)
- Reading Assignment: pp. 125-127 (start at bottom of p. 125), 133-134
- Reading Assignment: pp. 27-29, 36-38

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- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329
- Jonathan E Lopez on Reading Assignment: pp. 325-top 329

When is the domain always ( negative infinity and positive infinity)? And when shifting right and left in this section and section 1.3 do you shift either direction that number or are you shifting to that point on the graph? So if it said X-1 does that mean move right 1 or move right to point 1 on the X axis? Other than this question the rest of the section made sense! Section 1.3 was really confusing for me to read and know what was going on, but doing the homework in that section made more sense than reading what was going on. Was interesting to learn that the logX as the X increases the line gets closer to the x-axis pg 32 figure 21. I need help with composite function also.

I pretty much understand everything in the assigned reading except the transformation of functions. I always had trouble understanding how to shift certain functions and why they shifted in that specific direction.

The key with shifting is to notice that if you do f(x)+c or f(x)-c, you are adding or subtracting from a y-coordinate, which means you are shifting up or down.

If you do f(x+c) or f(x-c), you are adding or subtracting from the x-coordinate, so you will be shifting left or right. Here, there shifts are the opposite of what you expect. The reason is that if you are adding c, for example, the x-value required to produce a specific output is now lower, meaning the graph moves left. For example, if (5,10) is on the graph of f(x), then (3,10) is on the graph of f(x+2) (to get an input of 5, we only need x=3 since we are adding 2 to x). So the net effect is a shift left by 2 units.

The domain is (-\infty,\infty) when any number can be substituted into the function without a problem. For example, this is the domain of f(x)=x^2, since you can square any number. This is not the case with the f(x)=\sqrt{x}, because you cannot take the square root of a negative number. So for f(x)=\sqrt{x}, the domain is [0,\infty).

If f(x) is the “starting function”, then f(x-1) is a new function, and the graph of the new function is the graph of the old function (f(x)) shifted RIGHT by 1 unit. So you aren’t necessarily moving to a specific point, you are taking the entire graph and (in this case) sliding it 1 unit to the RIGHT.

The reading overall made sense and was review. The transformations all were explained well enough where it made sense as to which direction a function would be translated, and when it would be stretched or shrunk. Sometimes it is hard to visualize what the transformation would look like, such as in problem 17 on page 42 (1/2(1-cosx)) when the function’s transformations are not written in the clear way that the reading examples showed. When translating a trigonometric function right or left, does a value of 1 indicate sliding a whole pi? On page 28 example 4, it made sense how the equation for the falling ball was calculated, however I haven’t taken trig since sophomore year and might need some review on how to solve for least squares method and what that tells you in context of the question.

For the example y=1/2(1-cos(x)), the “starting function” to use is the graph of y=cos(x), since we know the shape of it. To transform its graph into the graph of y=1/2(1-cos(x)), we need to do three things:

1) Reflect over the x-axis to get the graph of y=-cos(x);

2) Shift 1 unit UP to get the graph of y=1-cos(x) (note that this is the same as y=-cos(x)+1, so we are adding 1 to the previous function); and

3) Shrink vertically by a factor of 1/2 to get y=1/2(1-cos(x)). The shrinking is hard to visualize because different points are generally “moved” by different amounts.

As for Example 4 on page 28, the goal is to come up with a function that models the height of the ball as a function of time. Based on the data that was collected, the points were plotted and had a parabola-like shape. That means a quadratic function is a good choice to model the height of the ball. They use calculator or computer to come up with the actual function. Don’t worry about actually coming up with the function – as long as you see that a quadratic function “fits” the data, and understand how to figure out when the ball hits the ground, that is sufficient.

So when reading the pages assigned for this section, I came to the section on polynomials me question is whether there is uses of polynomials in the degree of 4&5 is used. Because in the book it sorta skims of those two degrees and just shows a sample graph in Figure 8. Also for trigonometric functions will the range always be in the interval of that has the positive and negative of the same number, like [-1,1] or [-9,9] or will it change and have different numbers? Then also in the section about polynomials where a = n and n is a positive integer, will the graphs with positive n or negative n have similar shapes.

Yes, polynomials with higher degrees often come up when modeling “real-world” situations. The reason the book generally focuses on lower degree polynomials, like degree 2 and degree 3, is that they are much simpler to work with. For example, the quadratic formula lets us solve any quadratic equation; there is no formula to do that in the case of a degree 4 or degree 5 equation.

The domain and range for trig functions vary based on the function. It’s true that the range of sin(x) and cos(x) is [-1,1], but the range for tan(x) is (-\infty,\infty). The domain of sin(x) and cos(x) is all real numbers, whereas the domain for tan(x) is all real numbers except odd multiples of \pi/2.

For polynomials, the shape of the graph can vary greatly based on the terms and the coefficients that the polynomial has. For power functions, f(x)=x^n, the shape will be similar whenever n is even (“u-shaped”) and similar whenever n is odd and at least 3.

If n is negative, the graph will have a “hyperbola”-like shape, similar to the graph on the top of page 30. The reason is that the function is undefined when x=0. For example, f(x)=x^{-2}=1/(x^2), which is undefined when x=0.

The section on polynomials was pretty straight forward, apart from a few small things. In example 4 of the polynomial section, when they say they used a graphing calculator to obtain their quadratic model, I was confused with how they did that. For the power functions, should we have a general idea of what the graph will look like just from looking at the equation? Also, should we memorize all transformations of functions, and what each transformation does to the function? The remaining of the examples for transforming functions seems straightforward, and looks like you have to just recognize the translation and remember what it does to the function.

The first step is to plot the data points (Figure 9 on page 28). Since the points appear to have a parabola-like shape, a quadratic function will fit the data fairly well. As far as how to get the actual quadratic function, just knowing that a calculator or a computer will get it is sufficient. This if often what happens in practice: data are collected, points are plotted, based on the pattern of the points a certain type of function is determined to fit the data best, and a computer is used to obtain the actual function.

For power functions, you should know the general shape of the graph based on the equation. For example, x^2, x^4, x^6, etc. all have a “u-shape”; x^3, x^5, x^7, etc. all have a similar shape; x^{1/2}, x^{1/4}, x^{1/6}, etc. all have a similar shape; and x^{1/3}, x^{1/5}, x^{1/7}, etc. all have a similar shape.

As far as transformations, you should know what they do to graphs. The third question on today’s quiz is a typical type of question about transformations.

Polynomials were easy to understand and I understood that reading. The translations were a bit confusing however. For example, why would f(x)+c move the function to the left on a graph? Is the x-value not increasing? (Figure 1 on page 36)

So f(x)+c would actually move the graph up by c units. I think you may mean f(x+c). The reason the graph moves to the left is that the x-value required to produce a certain output is less than it is without the c.

For example, if f(x)=x^2, then x=4 produces an output of x=16. But for f(x+2)=(x+2)^2, to produce that output of 16, we now only need x=2. So the point (4,16) on the original graph slides to (2,16) on the new graph. So it moves 2 units left.

This is still review, so a majority of this is simple to understand, but good to look over again to refresh. I am a little confused of horizontal shrinking/stretching. It seems as though y=f(x/c) should shrink the graph rather than stretch it. I’m sure we will go over this in class tomorrow.

With transformations, when you modify the x-value itself, the result is typically the “opposite” of what you would expect. See the comment above for an example of why f(x+c) results in a shift to the LEFT.

For stretching/shrinking, we can use f(x)=x^2 again as an example. In this case, x=6 produces an output of 36. For the function f(x/2)=(x/2)^2, to produce that output of 36, we need x/2=6. So x has to be 12. Since you are dividing the x-value by a number before squaring, the value needs to be bigger to produce the same output as the original function. Thus, a horizontal stretch occurs.

On page 28, you learn that polynomials are used to model quantities that occur in the natural and social sciences. On page 29, you learn about different kinds of functions, such as power functions and root functions, and how their graphs look. On page 38, you kearn how to sketch the graph of functions by learning about what makes the function shift right, left, up or down, and also about what stretches them and reflects them. All of this is just basic review of concepts that most of us have heard before.

I thought the review of the polynomials was helpful, however example 4 as to why the quadratic formula was used. Also in the root functions why is the graph of the cube root of 3 go into the negatives why just the square root does not (like in figure 13 on page 29)? I thought that the transformations of functions were very straight forward, although I think I need to refresh my mind on the trig graphs.

In Example 4, the goal is to figure out the time at which the ball hits the ground. In other words, what t-value(s) produces a height of h=0 (ball is on ground when height is 0).

So once they have the quadratic model that fits the data, h(t), solving h(t)=0 will give those times. The quadratic formula is the best way to solve the equation in this case.

For even root functions (square root, fourth root, etc.), the domain is [0,\infty). This is why you don’t see the graph extend to the left of the y-axis. Remember that you can’t do an even root of a negative number.

For odd root functions (cube root, fifth root, etc.), the domain is all real numbers, which is why the graph extends indefinitely in both the left and right directions. It is okay to take cube roots of negative numbers. For example, the cube root of -8 is -2, so you will see a point (-8,-2) on the graph of the cube root function.

Acting as a review chapter, I have not had many issues with the content from chapter 1. Any concepts I struggled with through high school, I have come to understand upon reviewing them this semester. The first set of pages (27-29) gave a very clear depiction of polynomial and power functions. I always found these particular types of problems to be easy. Concepts I did struggle with a bit in high school were the graph of tan(x) and logarithmic functions. I decided to review these two concept on page 32, and the textbook helped to clear up a couple of misconceptions that I had. In the second set of pages (36-38), I found the table at the top of page 37 to be very helpful. I still find the concepts of vertical/horizontal shifts/stretches to be difficult, so clarifying the differences again in class would be very helpful.

For shifts/stretches, you can think of it like this:

If you are adding or subtracting directly to x, the result will be a horizontal shift, in the opposite direction expected (adding means LEFT, subtracting means RIGHT).

If you are multiplying or dividing directly to x, the result will be a horizontal stretch or shrink, again of opposite type as to what you would expect (multiplying means SHRINK, dividing means STRETCH).

If you are adding or subtracting to the entire function, the result is a vertical shift (adding means UP, subtracting means DOWN).

If you are multiplying or dividing to the entire function, the result is a vertical stretch or shrink (multiplying means STRETCH, dividing means SHRINK).

Reading over the polynomials pages it really helped that they provided graphs to illustrate and give a visual representation. I had learned about polynomials in high school so reading over this was good review for me. The section on transformations of functions was also review for me. I remembered the vertical and horizontal shifts but cannot completely remember the vertical and horizontal stretching. There are many rules for that but again having the graphs of pictures provided really helped. Figure 3 really helped me with the concept of the stretching and reflecting.

The polynomial section really helped me with understanding which way the polynomials are plotted on the graph. That way I am able to distinguish the difference between linear function and cubic function. Along with the way each different one of those functions lay out on the graph. Power functions have many different ways of plotting out on the graph. This section was a bit confusing to me, but after hearing your lecture today it cleared it up well. Transformations were easy for me to understanding but the stretching was unfamiliar to me, but I will see how I feel after class tomorrow.

I preferred the graphs in figure 4 on page 37 to the graphs in figure 3. The rules for changing the graphs made more sense when they weren’t piled on top of each other the way they are in Fig. 3. A few rules do seem to go counter to what you would expect. Such as f(x+c) shifting the graph to the left and f(cx) shrinking the graph instead of stretching it.

Yes, when you “modify” x directly as in f(x+c), f(x-c), f(x/c), and f(cx), the result is the opposite of what you would expect.

Example 4 on page 28 is very familiar to me because I have seen and done it before but, I completely forgot the quadratic formula until I reread this section. I always had a problem with “completing the square”, is there a way to remember it? Transforming functions is one of the easiest things to do. The graph cant really get messed up because it is exactly or similar to the original.

Yes, there is a formula for completing the square in general.

In the case of y=x^2+bx+c, which is what comes up on page 38, the way to complete the square is to do y=(x+b/2)^2+(c-b^2/4). This probably won’t come up all that often.

Everything in the reading was pretty simple to understand. I was a bit confused about which direction the graph shifts when using the g(x) functions. the boxes with the different rules really helped me to better understand the concepts.

g(x) is just the name given to the new function. For example, if the original function is called f(x), then a new function g(x) can be created via g(x)=f(x)+c. So if f(x)=x^2 and c=5, for example, then the new function is g(x)=x^2+5. Remember that f and g are just the names of the functions. Since the name f applies to the original function, the new function has to be given a different name. In this section, g is used for the new name.

I understand the concept that any polynomial is going to have the domain of all real numbers as long as the leading coefficient an is not equal to 0.Section 1.3 on Transformations of Functions is pretty much review because the transformation depends on how the c is applied to the function from within the parenthesis or outside of it as well as if it’s addition of subtration

Reading pages 27-29 helped me refresh my mind with polynomials. The figures throughout it helped me visually see and memorize how the graphs are suppose to look and the figures helped me further understand the content. Section 1.3 confused me a bit. For example figure 1 shows the translations of f but i don’t understand how y=f(x-c) moves to the right and y=f(x+c) moves to the left. I would think the opposite. Besides that the content section 1.3 was straight forward and the boxes of infromation helped me memorize the information that the paragraphs explained .

When you add or subtract a number directly to or from x, the transformation is the opposite of what you would expect.

For example, think of the function f(x)=x^2. When x=2, the output is 4. So (2,4) is a point on the graph.

Now think of the function f(x+2)=(x+2)^2. How do we get an output of 4 (like above)? In this function, notice that we no longer need x=2 to get 4 as the output – since we’re adding 2 to x before squaring, we only need x=0 to produce an output of 4. So the point (0,4) is on the graph of this new function.

So we can think of the point (2,4) on the original graph as moving to (0,4) on the new graph. So it has moved 2 units to the LEFT. So f(x+2) is the graph of f(x) shift LEFT by 2 units.

In the grand scheme of things functions are a bit confusing for me because I don’t understand domain in some context. For example if we have a cube root is the domain “all real numbers”? 1.3 discusses translations of functions which means that the graph will translate across the graph. Whether thats up or down, left or right. Each type of function is pretty self explanatory, for instance trig functions have some sort of trig sign such as sin or cos. Stretching and reflecting is pretty good too. If you think in terms of “x” or F(x) it makes a lot of sense whether you add or negate to figure out what will happen to your graph. If theres anything I don’t understand it’s trig, I need a brief overview of all trig functions because I can’t recall them from high school. Also this is a bit off topic but I really don’t know how to write in interval notation I went to the tutoring center and they tried to explain. I’m just not sure when to use brackets or parentheses.

Yes, the domain of the cube root function is all real numbers. This means you can take the cube root of any number. If you graph the cube root function, you will see that it extends forever in both the left and right directions, another indication that the domain is all real numbers.

For trig functions, we unfortunately aren’t able to talk very much about them in class. Appendix D in the textbook provides a fairly comprehensive overview.

As for interval notation, you use a closed bracket when the endpoint is INCLUDED in the interval; you use an open parentheses when the endpoint is NOT INCLUDED in the interval.

For example, if we say “all numbers greater than 2”, the interval would be (2,\infty). Note that 2 is not part of the interval because 2 is not bigger than 2. The open parentheses on the left means the number 2 is NOT INCLUDED.

If we say “all numbers greater than or equal to 2”, then the interval is [2,\infty). The closed bracket means the number 2 is part of the interval, since 2 is greater than or equal to 2.

On page 27 it says that a polynomial of degree 1 is in the form of P(x)= mx+b and the degree 2 is in the form of P(x)= ax^2 + bx + c. So would degree 3 be P(x)= ax^3 + ax^2 + bx + c? Everything else is pretty basic as far as the graphs and what happens when you add or subtract or multiply the function or coefficient (translations). The trig graphs were a little confusing though.

You are close with degree 3, except each coefficient should be a different letter. So if you were writing a degree 3 polynomial, you would write something like

P(x)=ax^3+bx^2+cx+d.

Unfortunately we don’t have much time to go over trig functions. I would focus on knowing the graphs of sine and cosine. The graph of the tangent function may come up in some homework problems. With that, knowing that tan(x)=sin(x)/cos(x) is helpful because you know that tan(x) is undefined when cos(x)=0 (which happens at odd multiples of pi/2). In the text, you will see dotted lines at each of the places in the graph of the tangent function.

I wouldn’t worry about the graphs of the other trig functions (secant, cosecant, and cotangent). We don’t use these very often.