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) all real numbers; all real numbers; all real numbers

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(2) Find all $x$- and $y$-intercepts.
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(2)
$$
x_{1}=\frac{-2\sqrt{3}}{3}, x_{2}=\frac{2\sqrt{3}}{3};
$$

$$
x_{1}={-2}, x_{2}={2};
$$

$$
x_{1}=(-2\sqrt{2}), x_{2}=(2\sqrt{2})
$$

no y intercepts

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(3) Find all equations of horizontal asymptotes.
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(3)$$
y=0;
y=0;
y=0;
$$

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(4) Find all equations of vertical asymptotes.
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(4) $$
x=0;
x=0;
x=0;
$$

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(5) Find the interval(s) where $f$ is increasing.
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(5)

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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) $$
f\left(2\right)=4;
$$

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there is none
$$

$$
f\left(\frac{2\sqrt{30}}{3}\right)=\frac{9\sqrt{30}}{250}
$$

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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)$$
f\left(-2\right)=-4;
$$

$$
f(2\sqrt{2})=-\frac{3}{4};f(-2\sqrt{2})=-\frac{3}{4};
$$

$$
f\left(-\frac{2\sqrt{30}}{3}\right)=-\frac{9\sqrt{30}}{250}
$$

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(8) Find the interval(s) on which the graph is concave downward.
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(8)

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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)

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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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