Given the graph I'm asked for the degree.
I can see how it is odd. I can see how the $a$ value is negative.
I can see the inflection point at $0$.
My question is how would I know if this is degree 3 or 5 or 7....?
The book's answer is this is degree 5.
It also says it starts in quadrant 1, and from my trig background, I'd say it starts (left to right) in quadrant 2.
But my main question is how is it degree 5 vs degree 3 or degree 7?
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$\begingroup$Just to reiterate the comments made above, the question itself is flawed. An image of only a finite section of the graph of a function is not enough information to conclude much of anything except for lower bounds on degrees.
Here are a few graphs which look very similar to the original graph in the question. Note, these are all graphs of different functions! I did not just copy the same image three times to make a point!
$$y=-\frac{x^5}{20}+\frac{2x^3}{3}$$
This is probably the function used to generate the image in the textbook, however look at these others...
$$y=\frac{x^7}{10^{10}}-\frac{x^5}{20}+\frac{2x^3}{3}$$
Ironically, this graph will have the opposite horizontal limits as the original graph. In fact, as $x\to-\infty$ this graph will become very negative! not positive! This is despite what we can see from the image.
One more:
$$y=\frac{x^6}{10^{10}}-\frac{x^5}{20}+\frac{2x^3}{3}$$
Here in this picture, the polynomial isn't even of odd degree, but rather is even.
What we can determine is that since the number of inflection points is visibly greater than or equal to $3$, the degree of the polynomial must be at a minimum $5$ but could be more than that too. We know this because inflection points occur when the second derivative is equal to zero. As such we know that the second derivative of the polynomial in question is degree at least three. By integrating twice we get the original function must be a polynomial of degree at least five.
Any further information beyond this cannot be gleaned from the picture alone. The correct answer to the originally phrased question is:
Is the function even or odd: unknown (but appears odd from the given information as any image of a function should include as much relevant information as possible)
What is the degree of the function: at least five (it appears to be five but it could be more)
Count the number of inflection points. That rules out degree $3$ (a third degree polynomial can only have one inflection point). As for degree $7$, there is no way to really tell from the graph that there isn't a $10^{-100}x^7$ term in there (or a $10^{-100}x^6$ for that matter, which would ruin the odd-ness), so if it truly is degree $5$, then that's not something we can see from a picture like this.
$\endgroup$ 2 $\begingroup$You can see two roots at approximately $x = 4$ and $x = -4$. Plus, there is a root of multiplicity $3$ at $x = 0$. Therefore it is an equation of degree $5$.
It's actually not possible to determine exactly what the degree of the equation actually is from a graph. But $5$ is the answer they're looking for.
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