---
# System prepended metadata

title: 'Math 181 Miniproject 7: The Shape of a Graph'

---

Math 181 Miniproject 7: The Shape of a Graph.md
---
---
tags: MATH 181
---

Math 181 Miniproject 7: The Shape of a Graph
===

**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 $3.1$. 

---

:::info
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. 
:::

(1)
$(-\infty,0]$ and $[0,\infty)$
:::info
(2) Find all $x$- and $y$-intercepts. 
:::

(2)
$\frac{12x^2-16}{x^3}=0$
$12x^2=16$
$x^2=16/12$
$x^2=4/3$
$x= + or - \sqrt(\frac{4}{3})$
$y=\frac{12(0)^2-16}{(0)^3}$
$y=-16/0$
y is undefined


:::info
(3) Find all equations of horizontal asymptotes. 
:::

(3)
y=0 since f(x) never touches 0 at y
:::info
(4) Find all equations of vertical asymptotes. 
:::

(4)
x=0 since f(x) never touches 0 at x 
:::info
(5) Find the interval(s) where $f$ is increasing.
:::

(5)
$\frac{-12(x^2-4)}{x^4}=0$
$-12(x^2-4)=0$
$x^2-4=0$
$x^2=4$
$x=+ or - 2$
:::info
(6) Find the $x$-value(s) of all local maxima. (Find exact values, and not decimal representations) 
:::

(6)
$f'(x)=-\frac{12(x^2-4)}{x^4}$
f'(1)>0
f'(3)<0
x=2 local maxima
:::info
(7) Find the $x$-value(s) of all local minima. (Find exact values, and not decimal representations)
:::

(7)
$f'(x)=-\frac{12(x^2-4)}{x^4}$
f'(-3)<0
f'(-1)>0
x=-2 local minima
:::info
(8) Find the interval(s) on which the graph is concave downward. 
:::

(8)
 $f''(x)=\frac{24(x^2-8)}{x^5}$
 $24(x^2-8)=0$
$x^2=8$
$x=+ or - \sqrt{8}$
f''($\sqrt{8}$)= (+) and concave up
f''($-\sqrt{8}$)= (-) and concave down

:::info
(9) State the $x$-value(s) of all inflection points. (Find exact values, and not decimal representations)
:::

(9)
$f''(x)=\frac{24(x^2-8)}{x^5}$
$\frac{24(x^2-8)}{x^5}=0$
$24(x^2-8)=0$
$x^2=8$
$x=+ or - \sqrt{8}$
f''(-3)<0
f''($-\sqrt{8}$) inflection point
f''(-1)>0
f''(1)<0
f''($\sqrt{8}$) inflection point
f''(3)>0


:::info
(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.
:::

(10)
![](https://i.imgur.com/NNxNlkt.png)






---

To submit this assignment click on the Publish button ![Publish button icon](https://i.imgur.com/Qk7vi9V.png). Then copy the url of the final document and submit it in Canvas.



