How do you know a quadratic would be appropriate to use when modeling a real-world solution?

Question

I know this sounds like a simple question, but it was a question my sister had on her homework (Algebra 2) that I had a hard time answering and explaining why she was told that her answers were wrong. Here are the two questions:

1a. How do you know a quadratic would be appropriate to use when modeling a real-world solution?

1b. Give two specific examples of when a quadratic can be used?

Her answer for part a:

"When the situation has a parabola" which was marked wrong with a note from the teacher saying "such as?"

Her answer for part b:

1. To find the parabola of an equation
2. To factor / complete the square

which was marked wrong with simply a "?"

I argued that for part b, the teacher was likely looking for real world examples (e.g. the height of a ball that is kicked over time, or the path a person takes when jumping off of a cliff into water). For part a, I was less sure, as my sister argued that the teacher writing "such as?" makes the question the same as part b, which I understand her argument for. I said I think that part a is more about how you know something is parabolic, and I proposed the idea that the path would be symmetrical, but this applies also to other functions, and does not properly represent the cliff jumping scenario, like it does the ball example. Can someone help provide a better answer for part a, and help to distinguish what is being asked between part a and b?

Edit: I wanted to note that the question does not give any details or examples to consider, so it is not enough to say that you can take data and check the second differences or the R^2 value. I imagine it as reading a word problem - what about that word problem tells you that it can be represented quadratically?

Answer 1

My interpretation of these questions is as follows:

A)

Q: How do you know that a dataset can be represented by a quadratic?

A: If the second difference of the data is constant than the data is parabolic. For example, in the values,

$(x,y), (1,1), (2,4), (3,9)$

The second difference, 2, is constant.

OR:

If a calculator is allowed, a quadratic regression with an $\Bbb R^2$ close to $1$ would indicate that the values can be accurately represented with a parabola.

B)

What are real world examples of quadratics?

A: Some real-world examples of quadratics include d-t graphs of a ball being thrown straight up in the air, projectile motion, and finance.

Answer 2

For part b) here's an interesting example applicable to the real world.

The intensity (i.e. brightness) of light behaves in a way described by the "inverse-square law" that is to say,

$$\text{intensity of light at a point} \propto \frac{1}{\text{distance of that point from the light source}^{{\color{red} 2}}}$$

where the symbol $\propto$ means proportional to. Notice the square in this formula, making it a quadratic equation.

(If it does not immediately look like a quadratic equation due to the division by the squared term, note that, for example, if $a = \frac{1}{x^2}$ then $a \cdot x^2 = \frac{1}{x^2} \cdot x^2$ (by multiplying both sides by $x^2$) giving $ax^2=1$, i.e. $ax^2-1=0$, which is certainly a quadratic equation.)

A nice example of this in a real world application: I heard a certain anecdote, wherein someone who worked on a certain animated kids movie (I forget which one exactly) said that in order to render the light in the movie, the quadratic equation must have been used millions if not billions of times (by a computer of course, but the quadratic equation must have been programmed in initially at least once by someone).

From Wikipedia, an illustration depicting the inverse-square law:

Notice how at a distance of $2r$, there are $2^2 = 4$ squares that the light has been spread out onto, and at a distance of $3r$, there are $3^2 =9$.