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Algebra Ch2.4 Set IV

Question from Cynthia:
Answer from Dr. Callahan:
You are correct. I assume you have a blue textbook and this is on page 72. It appears the problem got updated between versions and the solutions manual did not.
The y-axis is the weight of the contestant in pounds after he (she) eats each hot dog.
Then I can see in #2 that each hot dog eaten increases their weight in a direct variation.
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Algebra 1 Ch2.5 #8c


Why does the perimeter not vary directly with it’s length (when the length changes)

Answer from Dr. Callahan:

If you look at the formula for perimeter is becomes

P = 2x + 6

We see it is linear,

y = ax + b

but a direct variation is a subset of linear where the b =0 (see page 85)

This is a minor definition you will rarely see – so from now on just expect Linear or not.


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Algebra 1 Ch2.5 #7

Question from Nathan:

Lesson 5 Chapter 2 page 93 number 7.  I do not understand the answer to a and b and my mom cannot explain it to me.  Difference to me is subtraction not multiplication and the answer said it is the difference between successive values of Y on a.

Answer from Cassidy Cash:

Yes, “difference” does mean subtraction. Notice that all of the y values are 3 more than the one before.

8-5 = 3

11-8 = 3

14-11 = 3

And it would be that way on throughout the table, were you to extend the x values past three. That is what it means by “the difference between successive values of y”, meaning that if you take each y value and subtract it from the y-value following it, then you will get 3.

Letter b is very very basic, in that it just wants you to recognize that the 5 is paired up with the 0. It is teaching you to understand that the y-values are determined by whatever the x value was for that point. Since x was zero, y was 5. That is all it is asking for that one.

Does this help? God Bless, Cassidy Cash

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Algebra 1 Ch2.3 Function & Formulas

Question from Jake:

We have a question on functions and the fact that they don’t always have a formula. Would this be an example of a function without a formula? Say, for example that there was a function for rain, and the function was almost completely random, 2 inches one month, 4 inches the next, 10 inches the following, and 1 inch the last month measured. This graph is completely random. The amount of rain could be based on dozens of variables, i.e. rain patterns, amount of moisture in the air, etc. A graph like this could also go for stocks, storms, etc. Would these all be examples of functions with no formula?

Thanks a lot

Answer from Dr. Callahan:

Yes – what you have described in your examples are what we call random functions – since we often MODEL these relationships based on random processes. Fact is, they are NOT random – but the massive amount of data needed to build a formula makes it not worth the trouble.
But either way – they are functions that do not have formulas.


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Algebra Chapter 2 Lesson 4 – Direct Variation


Hello. I am a homeschooling mom using your DVDs and the Jacobs text. I also happen to have a bachelors in math and taught Algebra in VA for 5 years. (But that was 14 years ago now.) In the Set I questions we are sometimes asked review type questions about functions. Sometimes we are asked “What type of function is this?” How do we know when to answer “direct variation” and when to answer “linear function.” I do know the difference between them, but not sure when a linear function IS a direct variation, which answer is correct. Hope my question is not too confusing…Thanks so much!


Direct Variation equations are also linear equations, so understandably they get overlapped. The distinction is the y-intercept. Direct Variation lines will always go through the origin Their y-intercept will always be 0, that’s why their equations always fit the form of y = ax. It is the same as a line which you have y = mx + b, but the b term is 0 for direct variation, so it shortens to y = mx (or y = ax, which is the same thing).

When you are answering the questions in the book, a lot depends on what you are studying in the text, or that section as to whether the book is looking for direct variation or linear. The distinction, though, is whether or not it goes through the origin. If you have an equation that is both a linear function, but also a direct variation equation, I would allow both answers as correct. If you want to be technical about what’s correct, then the answer to “what kind of function is this” would probably be “linear function with a direct variation equation” since direct variation is a special case of linear function. But take into context what the student is learning. If they have already learned what linear equations are, and are now in a lesson specifically designed to teach them about direct variation, then when they come across a problem that graphs as a line through the origin, then the likely answer is “direct variation”.

Does that help?

God Bless,

Cassidy Cash


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Algebra Chapter 2 Graphing


Hi, I am going through the Algebra one course and am struggling through Chapter 2 (all of it!)and especiallythe graphs. My mom is super picky and it has to be PERFECT. Is there a short way to do it without a ruler or graph paper? Also, how exactly do you do decimals on graphs? Thanks.



Here is a video that goes over some tips about how to graph accurately without using graph paper. They should help!



However, if you are struggling with the whole chapter, I would be happy to help you with some particular problems. Take a look at the latest lesson you’ve been working in, and pick one or two problems that are particularly difficult. Then email me the lesson and problem numbers, and I can go over graphing as it applies to those problems specifically. I hope this helps!