---
# 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)
The domain of $f(x )$ is 
$(-∞,0)and (0,∞)$


:::info
(2) Find all $x$- and $y$-intercepts. 
:::

(2)
$f\left(x\right)=-\frac{12\left(x^{2}-4\right)}{x^{4}}$
Set the equation to $0$
$f\left(x\right)=-\frac{12\left(x^{2}-4\right)}{x^{4}}=0$
$0=x^2-4$
$x=±2$
$x^4=0$


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

(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, $y=0$

:::info
(4) Find all equations of vertical asymptotes. 
:::

(4)
$f'\left(x\right)=\frac{12\left(x^{2}-4\right)}{x^{4}}$
The vertical asymptotes is $x=0$
:::info
(5) Find the interval(s) where $f$ is increasing.
:::

(5) f is increasing at (-∞,-2) and (2,∞)

![](https://i.imgur.com/tnGF25X.png)


  

:::info
(6) Find the $x$-value(s) of all local maxima. (Find exact values, and not decimal representations) 
:::

(6) From the first derivative.
Based on the number line f has a local max of $f(-2)$.
![](https://i.imgur.com/j2KfOg8.png)



:::info
(7) Find the $x$-value(s) of all local minima. (Find exact values, and not decimal representations)
:::

(7)Based on the number line f has a local min of $f(2)$
![](https://i.imgur.com/6VRomtk.png)

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

(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:
$x=-2\sqrt{ 2},  2\sqrt{ 2}$
![](https://i.imgur.com/tNYPjpe.png)
So, f is concave down on the intervals $(-∞,-2\sqrt{ 2})$ and $(0, 2\sqrt{ 2})$



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

(9)
f has an inflection point at $x= ±2\sqrt{ 2}$
![](https://i.imgur.com/y8AlNF1.png)

:::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/UViWP25.png)
![](https://i.imgur.com/cXh8HfV.png)
![](https://i.imgur.com/Zm5uCve.png)
![](https://i.imgur.com/9qeImYu.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.



