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Step-by-step explanation:
<span>For this case we have the following functions transformation:
Vertical expansions:
To graph y = a * f (x)
If a> 1, the graph of y = f (x) is expanded vertically by a factor a.
f1 (x) = (3x) ^ 2
Horizontal translations
Suppose that h> 0
To graph y = f (x-h), move the graph of h units to the right.
f2 (x) = (3x-6) ^ 2
Vertical translations
Suppose that k> 0
To graph y = f (x) + k, move the graph of k units up.
g (x) = (3x-6) ^ 2 + 3
Answer:
expanded horizontally by a factor of 3, horizontal shift rith 6, vertical shift up 3</span>
1.
has a horizontal asymptote at ![y=-4](https://tex.z-dn.net/?f=y%3D-4)
This means that
![\displaystyle\lim_{x\to\pm\infty}f(x)-(-4)=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bx%5Cto%5Cpm%5Cinfty%7Df%28x%29-%28-4%29%3D0)
(for at least one of these limits)
2.
has a vertical asymptote at ![x=3](https://tex.z-dn.net/?f=x%3D3)
This means that
has a non-removable discontinuity at
. Since
is some rational function, there must be a factor of
in its denominator.
3.
has an
-intercept at (1, 0)
This means
.
(a) With
![f(x)=\dfrac{ax+b}{x+c}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac%7Bax%2Bb%7D%7Bx%2Bc%7D)
the second point above suggests
. The first point tells us that
![\displaystyle\lim_{x\to\pm\infty}\frac{ax+b}{x-3}+4=0=\lim_{x\to\pm\infty}\frac{ax+b+4x-3}{x-3}=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bx%5Cto%5Cpm%5Cinfty%7D%5Cfrac%7Bax%2Bb%7D%7Bx-3%7D%2B4%3D0%3D%5Clim_%7Bx%5Cto%5Cpm%5Cinfty%7D%5Cfrac%7Bax%2Bb%2B4x-3%7D%7Bx-3%7D%3D0)
In order for the limit to be 0, the denominator's degree should exceed the numerator's degree; the only way for this to happen is if
so that the linear terms vanish.
The third point tells us that
![f(1)=\dfrac{a+b}{1-3}=0\implies a=-b\implies b=4](https://tex.z-dn.net/?f=f%281%29%3D%5Cdfrac%7Ba%2Bb%7D%7B1-3%7D%3D0%5Cimplies%20a%3D-b%5Cimplies%20b%3D4)
So
![f(x)=\dfrac{-4x+4}{x-3}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac%7B-4x%2B4%7D%7Bx-3%7D)
(b) Since
![f(x)=\dfrac{rx+s}{2x+t}=\dfrac12\dfrac{rx+s}{x+\frac t2}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac%7Brx%2Bs%7D%7B2x%2Bt%7D%3D%5Cdfrac12%5Cdfrac%7Brx%2Bs%7D%7Bx%2B%5Cfrac%20t2%7D)
we find that
, and
and
.
Answer:
1. 2 1/4
2. 8.25
3. 8 6/8
4. 51.18
5. 56.875
6. $6.22
7. 62.14
Step-by-step explanation: