Graphs of Functions

[mathdad]

1. Why does the graph of f(x) = 2x^2 + 3x - 4 open upward?

My Answer:

The coefficient of the x^2 term is positive. Is there another reason?

2. Why does the graph of f(x) = - 3x^2 + 6x + 5 open downward?

My Answer:

The coefficient of the x^2 term is less than 0. Is there another reason?

 

[tkhunny]

Sounds good to me.

 

[mathdad]

Sounds good to me.

I find this level of algebra quite fascinating, to say the least. I am currently in Chapter 4 in the Michael Sullivan College Algebra 9th Edition. It's a user-friendly textbook (like all his books). Each chapter is divided into several sections.

I am in Section 4.4 on my way to cover every chapter/section. My weekly time is limited due to full-time employment. So, I decided to answer no more than 20 questions per section.

I also MUST PASS the Chapter Test with 70 percent or 14/20 to proceed. If my test score is less than 70 percent, I simply repeat the chapter. I take my study time very seriously. It makes no sense to read through the pages of a math textbook like a magazine. What do you suggest in terms of my endeavor to "master" at least the high school math content?

 

[Steven G]

I find this level of algebra quite fascinating, to say the least. I am currently in Chapter 4 in the Michael Sullivan College Algebra 9th Edition. It's a user-friendly textbook (like all his books). Each chapter is divided into several sections.

I am in Section 4.4 on my way to cover every chapter/section. My weekly time is limited due to full-time employment. So, I decided to answer no more than 20 questions per section.

I also MUST PASS the Chapter Test with 70 percent or 14/20 to proceed. If my test score is less than 70 percent, I simply repeat the chapter. I take my study time very seriously. It makes no sense to read through the pages of a math textbook like a magazine. What do you suggest in terms of my endeavor to "master" at least the high school math content?

I think that you wanting to get 14/20 = 70% might be a mistake as 70% is not necessarily a good score. You should expect a 100% on the test! Now do you need to really get a 100%? No, not at all. If you made a careless mistake then that is ok.

Here is a quick story. When I was in graduate school my friend noticed something wrong that I was doing with basic arithmetic problems. In computing say 8+3 I would see 11 so quickly that I added 3 to it and get 14. Well I never made that type of mistake again.
Now lets go back to when I was in calculus. I had the hardest time maintaining an A average because of my silly arithmetic mistakes. Did this mean I was not an A student in Calculus. Of course not. In fact on all my exams I never made a calculus mistake--I sure made lots of arithmetic mistakes and a few silly algebra mistakes but I was an A calculus student.

What I am getting at is you need to look at your mistakes and see how severe they are and decide from there if you should move on.

When I first started teaching at a community college I felt that the graduation requirement was correct in that all students needing to pass algebra to graduate. Now I am confused about this. I see students who pass the course but never got a problem correct on any exam! They would make a mistake on every problem (usually severe) but with partial credit get decent grades (and I am a tough grader!). Did they learn any algebra? Well I guess yes, but not enough to do any problem! You do not want to be one of these students. Move on only after you master the section!

 

[Steven G]

I do not like that you, a beginning algebra student, say that the parabola opens upwards if a>0 and downwards because a<0.
I want you to know why! I tell my students never to believe anything their math teachers tell them until they can see it for themselves. I think that one reason some students are so weak in math is because they accepted too much as fact and it is too late for them to now try to understand. Seriously, never accept anything that is in your book as true until you see it on your own!

I remember one day in my linear algebra class my professor stated a theorem and I said to him that this just can't be true. He was very very upset with me and said that I just have to believe that what he said was true. I said no(!) that you have to prove it to me before I will believe it. He looked at me, thought for a moment and then proceeded to prove the theorem to me and the class. He knew that I was right!

So back to your problem. So after plotting points for many many quadratic equations we see that there are two different possible shapes.
Now what happens when x is an extremely large positive value or an extremely large negative number if a>0 and if a<0. When you see this I will accept your original answer!

 

[mathdad]

I think that you wanting to get 14/20 = 70% might be a mistake as 70% is not necessarily a good score. You should expect a 100% on the test! Now do you need to really get a 100%? No, not at all. If you made a careless mistake then that is ok.

Here is a quick story. When I was in graduate school my friend noticed something wrong that I was doing with basic arithmetic problems. In computing say 8+3 I would see 11 so quickly that I added 3 to it and get 14. Well I never made that type of mistake again.
Now lets go back to when I was in calculus. I had the hardest time maintaining an A average because of my silly arithmetic mistakes. Did this mean I was not an A student in Calculus. Of course not. In fact on all my exams I never made a calculus mistake--I sure made lots of arithmetic mistakes and a few silly algebra mistakes but I was an A calculus student.

What I am getting at is you need to look at your mistakes and see how severe they are and decide from there if you should move on.

When I first started teaching at a community college I felt that the graduation requirement was correct in that all students needing to pass algebra to graduate. Now I am confused about this. I see students who pass the course but never got a problem correct on any exam! They would make a mistake on every problem (usually severe) but with partial credit get decent grades (and I am a tough grader!). Did they learn any algebra? Well I guess yes, but not enough to do any problem! You do not want to be one of these students. Move on only after you master the section!

You said the following:

". . . you need to look at your mistakes and see how severe they are and decide from there if you should move on."

My computation errors are small, mostly sign errors. For example, instead of x = 4, I write x = - 4 but I understand the general problem, the big picture.

You also said:

"Move on only after you master the section!"

1. I work 40 overnight hours. This means I sleep during the day. Monday is my Friday as I work Thursday through Monday. Working overnight hours is not even remotely the same as working during the day and/or the late afternoon, early night shift. Giving up my sleep time takes its toll on the body.

2. The only FULL DAY I have to take of personal errands (and somehow squeeze math into my schedule) is Tuesday.

3. I am off Tuesday and Wednesday. However, I go to bed at 2pm to start my Thursday morning shift Wednesday at midnight from 12am to 8am. Remember, the next day begins at midnight.

4. I took off this past Sunday into Monday morning (my Friday) because I had a podiatrist appointment at the VA Hospital.

5. I wish I could "master" every chapter/section but this is an impossibility based on 1 through 4 above (and other reasons). In other words, life often gets in the way.

 

[mathdad]

I do not like that you, a beginning algebra student, say that the parabola opens upwards if a>0 and downwards because a<0.
I want you to know why! I tell my students never to believe anything their math teachers tell them until they can see it for themselves. I think that one reason some students are so weak in math is because they accepted too much as fact and it is too late for them to now try to understand. Seriously, never accept anything that is in your book as true until you see it on your own!

I remember one day in my linear algebra class my professor stated a theorem and I said to him that this just can't be true. He was very very upset with me and said that I just have to believe that what he said was true. I said no(!) that you have to prove it to me before I will believe it. He looked at me, thought for a moment and then proceeded to prove the theorem to me and the class. He knew that I was right!

So back to your problem. So after plotting points for many many quadratic equations we see that there are two different possible shapes.
Now what happens when x is an extremely large positive value or an extremely large negative number if a>0 and if a<0. When you see this I will accept your original answer!

1. I am not a formal student. I am not in college.

2. My college days ended in 1993. I graduated in June 1994.

3. I am not a beginning algebra student. Surely, I did not major in mathematics like many of the members here. I do not know math at your level but certainly know more than the average Joe. You would be shocked to learn that most people walking down the street have no idea what a linear equation is let alone solve something like 2x + 4 = 40.

4. This journey in math that I have decided to take is a review, a revisit of material learned over 2-3 decades ago.

 

[MarkFL]

Jomo is just asking you to think about what happens in the graph of a quadratic function when \(x\) tends either towards positive or negative infinity. The term containing \(x^2\) is going to dominate the other terms because it has the greatest exponent on the variable. And because this greatest exponent is even, the sign of \(x^2\) will be positive, and so it is the sign of the coefficient that ultimately determines the sign of the squared term.

Does that make sense?

 

[mathdad]

Jomo is just asking you to think about what happens in the graph of a quadratic function when \(x\) tends either towards positive or negative infinity. The term containing \(x^2\) is going to dominate the other terms because it has the greatest exponent on the variable. And because this greatest exponent is even, the sign of \(x^2\) will be positive, and so it is the sign of the coefficient that ultimately determines the sign of the squared term.

Does that make sense?

The sign of the coefficient of the x square term determines if the graph opens up or down. This is stated in my textbook.

 

[MarkFL]

The sign of the coefficient of the x square term determines if the graph opens up or down. This is stated in my textbook.

Understanding why this is true is the kind of thing that will help you, especially later on in your studies when you look need to consider function behavior as the independent variable grows without bound, or approaches a particular value.

 

[mathdad]

Understanding why this is true is the kind of thing that will help you, especially later on in your studies when you look need to consider function behavior as the independent variable grows without bound, or approaches a particular value.

I need to dig deeper into each chapter/section.

 

[Steven G]

You said the following:

". . . you need to look at your mistakes and see how severe they are and decide from there if you should move on."

My computation errors are small, mostly sign errors. For example, instead of x = 4, I write x = - 4 but I understand the general problem, the big picture.

You also said:

"Move on only after you master the section!"

1. I work 40 overnight hours. This means I sleep during the day. Monday is my Friday as I work Thursday through Monday. Working overnight hours is not even remotely the same as working during the day and/or the late afternoon, early night shift. Giving up my sleep time takes its toll on the body.

2. The only FULL DAY I have to take of personal errands (and somehow squeeze math into my schedule) is Tuesday.

3. I am off Tuesday and Wednesday. However, I go to bed at 2pm to start my Thursday morning shift Wednesday at midnight from 12am to 8am. Remember, the next day begins at midnight.

4. I took off this past Sunday into Monday morning (my Friday) because I had a podiatrist appointment at the VA Hospital.

5. I wish I could "master" every chapter/section but this is an impossibility based on 1 through 4 above (and other reasons). In other words, life often gets in the way.

Mathdad, you have something that most students of mathematics do not have. You have time! Fair enough you have limited time per week but unlike conventional students you get to study at your own pace. All I am saying is to take it slow and learn it well. What is the point for example to learn it at the level of a student who would earn a D in the course?
I see that you have some talent and the drive to learn math. This is excellent! Given a similar time constraint as yours I could see why a formal student with your talent might get a low grade in an algebra course. But you have unlimited time to learn the material so use it.

It is good that when you looked at your errors you realized your mistakes. So sure you can move onto the next section in this case. That is you have mastered the section. All I am saying is that when you feel that you did not master the topic then maybe you should not move on. I truly feel that only you know if you understand the material well so use your own judgement.

 

[mathdad]

Mathdad, you have something that most students of mathematics do not have. You have time! Fair enough you have limited time per week but unlike conventional students you get to study at your own pace. All I am saying is to take it slow and learn it well. What is the point for example to learn it at the level of a student who would earn a D in the course?
I see that you have some talent and the drive to learn math. This is excellent! Given a similar time constraint as yours I could see why a formal student with your talent might get a low grade in an algebra course. But you have unlimited time to learn the material so use it.

It is good that when you looked at your errors you realized your mistakes. So sure you can move onto the next section in this case. That is you have mastered the section. All I am saying is that when you feel that you did not master the topic then maybe you should not move on. I truly feel that only you know if you understand the material well so use your own judgement.

I agree. No argument at my end. I know exactly what I'm doing wrong. I am posting TOO MANY questions per day.

This is bad considering my limited time and lifestyle. For example, I have no PC and no laptop. All math work is done on my phone. From now on, I will not post more than 3 questions on days when I do have time for math review. I think 3 questions is more than enough to deeply step into the subject at hand. Thanks for your reply and motivation.