Math 181 Miniproject 7: The Shape of a Graph.md
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tags: MATH 181
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Math 181 Miniproject 7: The Shape of a Graph
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**Overview:** In this miniproject you will be using the techniques of calculus to find the behavior of a graph.

**Prerequisites:** The project draws heavily from the ideas of Chapter 1 and $2.8$ together with ideas and techniques of the first and second derivative tests from $3.1$.

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We are given the functions
$$
f(x)=\frac{12x^2-16}{x^3},\qquad f'(x)=-\frac{12(x^2-4)}{x^4},\qquad f''(x)=\frac{24(x^2-8)}{x^5}.
$$
The questions below are about the function $f(x)$. Answer parts (1) through (10) below. If the requested feature is missing, then explain why. Be sure to include the work/test that you used to rigorously reach your conclusion. It is not sufficient to refer to the graph.
(1) State the function's domain.
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(1)
The domain of $f(x )$ is
$(-∞,0)and (0,∞)$

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(2) Find all $x$- and $y$-intercepts.
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(2)
$f\left(x\right)=-\frac{12\left(x^{2}-4\right)}{x^{4}}$
Set the equation to $0$
$f\left(x\right)=-\frac{12\left(x^{2}-4\right)}{x^{4}}=0$
$0=x^2-4$
$x=±2$
$x^4=0$

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(3) Find all equations of horizontal asymptotes.
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(3)
$f'\left(x\right)=\frac{12\left(x^{2}-4\right)}{x^{4}}$
The rules of the horizontal asymptotes states that when the denominator is bigger than the numerator then, $y=0$

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(4) Find all equations of vertical asymptotes.
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(4)
$f'\left(x\right)=\frac{12\left(x^{2}-4\right)}{x^{4}}$
The vertical asymptotes is $x=0$
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(5) Find the interval(s) where $f$ is increasing.
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(5) f is increasing at (-∞,-2) and (2,∞)


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(6) Find the $x$-value(s) of all local maxima. (Find exact values, and not decimal representations)
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(6) From the first derivative.
Based on the number line f has a local max of $f(-2)$.

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(7) Find the $x$-value(s) of all local minima. (Find exact values, and not decimal representations)
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(7)Based on the number line f has a local min of $f(2)$

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(8) Find the interval(s) on which the graph is concave downward.
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(8)From the second derivative.
$f"\left(x\right)=\frac{24\left(x^{2}-8\right)}{x^{5}}$
Set the equation equal to 0.
$x^2-8=0$
Critical values are:
$x=-2\sqrt{ 2}, 2\sqrt{ 2}$
So, f is concave down on the intervals $(-∞,-2\sqrt{ 2})$ and $(0, 2\sqrt{ 2})$

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(9) State the $x$-value(s) of all inflection points. (Find exact values, and not decimal representations)
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(9)
f has an inflection point at $x= ±2\sqrt{ 2}$

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(10) Include a sketch of the graph of $y=f(x)$. Plot the different segments of the graph using the color code below.
* **blue:** $f'>0$ and $f''>0$
* **red:** $f'<0$ and $f''>0$
* **black:** $f'>0$ and $f''<0$
* **gold:** $f'<0$ and $f''<0$

(In Desmos you could restrict the plot $y=f(x)$ on the interval $[2,3]$ by typing $y=f(x)\{2\le x\le 3\}$.) Be sure to set the bounds on the graph so that the features of the graph that you listed above are easy to see.
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(10)

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