Check the picture below.
so the hyperbola looks more or less like so, is a hyperbola with a vertical traverse axis, with a = 4 and h = 0, k = 0.
![\bf \textit{hyperbolas, vertical traverse axis } \\\\ \cfrac{(y- k)^2}{ a^2}-\cfrac{(x- h)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2}\\ asymptotes\quad y= k\pm \cfrac{a}{b}(x- h) \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7Bhyperbolas%2C%20vertical%20traverse%20axis%20%7D%20%5C%5C%5C%5C%20%5Ccfrac%7B%28y-%20k%29%5E2%7D%7B%20a%5E2%7D-%5Ccfrac%7B%28x-%20h%29%5E2%7D%7B%20b%5E2%7D%3D1%20%5Cqquad%20%5Cbegin%7Bcases%7D%20center%5C%20%28%20h%2C%20k%29%5C%5C%20vertices%5C%20%28%20h%2C%20k%5Cpm%20a%29%5C%5C%20c%3D%5Ctextit%7Bdistance%20from%7D%5C%5C%20%5Cqquad%20%5Ctextit%7Bcenter%20to%20foci%7D%5C%5C%20%5Cqquad%20%5Csqrt%7B%20a%20%5E2%20%2B%20b%20%5E2%7D%5C%5C%20asymptotes%5Cquad%20y%3D%20k%5Cpm%20%5Ccfrac%7Ba%7D%7Bb%7D%28x-%20h%29%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D)
![\bf \stackrel{\textit{the asymptotes are}}{\pm\cfrac{2}{3}x}\implies \pm\cfrac{2}{3}x=k\pm \cfrac{a}{b}(x- h)~~ \begin{cases} h=0\\ k=0\\ a=4 \end{cases} \\\\\\ +\cfrac{2}{3}x=0+\cfrac{4}{b}(x-0)\implies \cfrac{2x}{3}=\cfrac{4x}{b}\implies 2bx=12x \\\\\\ b=\cfrac{12x}{2x}\implies b=6 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(y-0)^2}{4^2}-\cfrac{(x-0)^2}{6^2}=1\implies \cfrac{y^2}{16}-\cfrac{x^2}{36}=1](https://tex.z-dn.net/?f=%5Cbf%20%5Cstackrel%7B%5Ctextit%7Bthe%20asymptotes%20are%7D%7D%7B%5Cpm%5Ccfrac%7B2%7D%7B3%7Dx%7D%5Cimplies%20%5Cpm%5Ccfrac%7B2%7D%7B3%7Dx%3Dk%5Cpm%20%5Ccfrac%7Ba%7D%7Bb%7D%28x-%20h%29~~%20%5Cbegin%7Bcases%7D%20h%3D0%5C%5C%20k%3D0%5C%5C%20a%3D4%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5C%5C%20%2B%5Ccfrac%7B2%7D%7B3%7Dx%3D0%2B%5Ccfrac%7B4%7D%7Bb%7D%28x-0%29%5Cimplies%20%5Ccfrac%7B2x%7D%7B3%7D%3D%5Ccfrac%7B4x%7D%7Bb%7D%5Cimplies%202bx%3D12x%20%5C%5C%5C%5C%5C%5C%20b%3D%5Ccfrac%7B12x%7D%7B2x%7D%5Cimplies%20b%3D6%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Ccfrac%7B%28y-0%29%5E2%7D%7B4%5E2%7D-%5Ccfrac%7B%28x-0%29%5E2%7D%7B6%5E2%7D%3D1%5Cimplies%20%5Ccfrac%7By%5E2%7D%7B16%7D-%5Ccfrac%7Bx%5E2%7D%7B36%7D%3D1)
Answer:
For average sized artwork (about 11x14" to 30x20"), the common mat width is 2". If you artwork is smaller, you might consider a 1.75 or 1.5" mat.
Step-by-step explanation:
Answer:
28feet
Step-by-step explanation:
The height of the balloon is given as h(t) = -16t² + 40t + 12 at 2 seconds, the height is 28 feet
Answer:
a
4,5,8,11 are the domain
Step-by-step explanation:
x is always the domain, input and It could also be a function but I don't think that only four five eight and 11 are functions so I would go with a
