Forming equations from a given graph

[tiramisu589]

For each graph:

• Identify the graph as absolute value, quadratic, linear, even degree polynomial, odd degree polynomial, exponential growth, exponential decay, square root, cube root, piecewise, or linear inequality.
• Identify the zero(s) and asymptote(s), if any.
• Give an equation for each type of graph.*

*I need the most help with forming equations from a graph!

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1. 2. 3. 4.

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My work so far (may be incorrect!):

1. Exponential decay​

Zero(s): ?​

Asymptote(s): y = -2​

Equation: ?​

​

2. Exponential growth​

Zero(s): ?​

Asymptote(s): y = 4​

Equation: ?​

​

3. Odd degree polynomial​

Zero(s): ?​

Asymptote(s): none​

Equation: ?​

​

4. Odd degree polynomial​

Zero(s): x = -4​

Asymtote(s): none​

Equation: ?​

​

As you can see, I have a lot of blank spaces! My work may be incorrect, too. If you see any mistakes, I'd love if you point them out! Also, I think you need to know the equations first in order to find out the zeros in most of the above graphs (since the zeros aren't clear). I definitely need the most help with forming equations from a graph! Any help with this?

Thank you!

 

[Dr.Peterson]

The first two are translations (shifts) and reflections of the basic exponential function y = b^x. The asymptote tells you the vertical shift. You may be able to recognize what the base is; if not just write the function as, say, y = b^(x-a) + c, and plug in the points indicated, then solve for the parameters a, b, c.

The last two are cubics, so they can be written as y = ax^3 + bx^2 + cx + d. Again, one way to find the specific formula is to plug in four points to get four equations in the parameters, and solve for those. I'm not sure I see an easier way, but that way is not at all hard.

You're right that the zeros in most cases can only be determined once you have to equation -- and maybe not even then, for the case of the cubic, apart from an approximation.

But then, it just says, Give an equation for each type of graph. So maybe it doesn't even want the actual equation of the specific graph (though there is enough information to do so). I may have answered the question already!

 

[Steven G]

I rarely do this but I disagree with Dr P with the last two graph. Although the graphs may be cubic equations that was not one of the choices which the OP listed as possible answers. More importantly, the OP is correct in saying that it is an odd degree polynomial as any of the last 2 graphs can in fact be a higher (odd) degree than 3.

 

[Steven G]

For each graph:

• Identify the graph as absolute value, quadratic, linear, even degree polynomial, odd degree polynomial, exponential growth, exponential decay, square root, cube root, piecewise, or linear inequality.
• Identify the zero(s) and asymptote(s), if any.
• Give an equation for each type of graph.*

*I need the most help with forming equations from a graph!

-------

1. View attachment 12095 2.View attachment 12097 3. View attachment 12098 4. View attachment 12099

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My work so far (may be incorrect!):

1. Exponential decay​

Zero(s): ?​

Asymptote(s): y = -2​

Equation: ?​

​

2. Exponential growth​

Zero(s): ?​

Asymptote(s): y = 4​

Equation: ?​

​

3. Odd degree polynomial​

Zero(s): ?​

Asymptote(s): none​

Equation: ?​

​

4. Odd degree polynomial​

Zero(s): x = -4​

Asymtote(s): none​

Equation: ?​

​

As you can see, I have a lot of blank spaces! My work may be incorrect, too. If you see any mistakes, I'd love if you point them out! Also, I think you need to know the equations first in order to find out the zeros in most of the above graphs (since the zeros aren't clear). I definitely need the most help with forming equations from a graph! Any help with this?

Thank you!

The 2nd equation can not be exponential growth as the function is not growing. In fact it is constantly decreasing. What would you call such a function?

 

[Dr.Peterson]

I rarely do this but I disagree with Dr P with the last two graph. Although the graphs may be cubic equations that was not one of the choices which the OP listed as possible answers. More importantly, the OP is correct in saying that it is an odd degree polynomial as any of the last 2 graphs can in fact be a higher (odd) degree than 3.

An interesting point. And since we can't technically be sure it is a cubic rather than some other odd degree (or, really, any degree, since we don't know for sure what behavior is hidden outside the window of the graph!), it is impossible to answer the last question, whether it asks for the equation of this graph, or a general equation of this type of graph! The best you can do, without trusting that the function is what it looks like (I should have said it appears to be a cubic), is to answer, ax^(2n+1) + bx^(2n) + ... + c.

A horrible problem!

The 2nd equation can not be exponential growth as the function is not growing. In fact it is constantly decreasing. What would you call such a function?

Yes; I deliberately (implicitly) accepted the answer, because there is no other valid answer in the list, and no other way I can see to describe it without many more words. It is growth in the sense that its absolute value is growing, and that the coefficient on the exponent is positive.

Again, not a very good problem. Anything we say about it that doesn't criticize the author of the problem will be lacking something.