Q # 1
Explanation
Given the parabola
![f\left(x\right)=\left(x-3\right)^2-1](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D%5Cleft%28x-3%5Cright%29%5E2-1)
Openness
- It OPENS UP, as 'a=1' is positive.
Finding Vertex
The vertex of an up-down facing parabola of the form
![y=ax^2+bx+c\:\mathrm{is}\:x_v=-\frac{b}{2a}](https://tex.z-dn.net/?f=y%3Dax%5E2%2Bbx%2Bc%5C%3A%5Cmathrm%7Bis%7D%5C%3Ax_v%3D-%5Cfrac%7Bb%7D%7B2a%7D)
![\mathrm{Rewrite}\:y=\left(x-3\right)^2-1\:\mathrm{in\:the\:form}\:y=ax^2+bx+c](https://tex.z-dn.net/?f=%5Cmathrm%7BRewrite%7D%5C%3Ay%3D%5Cleft%28x-3%5Cright%29%5E2-1%5C%3A%5Cmathrm%7Bin%5C%3Athe%5C%3Aform%7D%5C%3Ay%3Dax%5E2%2Bbx%2Bc)
![y=x^2-6x+8](https://tex.z-dn.net/?f=y%3Dx%5E2-6x%2B8)
![a=1,\:b=-6,\:c=8](https://tex.z-dn.net/?f=a%3D1%2C%5C%3Ab%3D-6%2C%5C%3Ac%3D8)
![x_v=-\frac{\left(-6\right)}{2\cdot \:1}](https://tex.z-dn.net/?f=x_v%3D-%5Cfrac%7B%5Cleft%28-6%5Cright%29%7D%7B2%5Ccdot%20%5C%3A1%7D)
![x_v=3](https://tex.z-dn.net/?f=x_v%3D3)
Finding ![y_v](https://tex.z-dn.net/?f=y_v)
![y_v=3^2-6\cdot \:3+8](https://tex.z-dn.net/?f=y_v%3D3%5E2-6%5Ccdot%20%5C%3A3%2B8)
![y_v=-1](https://tex.z-dn.net/?f=y_v%3D-1)
So vertex is:
![\left(3,\:-1\right)](https://tex.z-dn.net/?f=%5Cleft%283%2C%5C%3A-1%5Cright%29)
Horizontal Translation
moves the graph RIGHT 3 units.
Vertical Translation
moves the graph DOWN 1 unit.
Stretch or Compress Vertically
As
, so it does not affect the stretchiness or compression.
Q # 2
Explanation:
![f\left(x\right)=-\left(x+1\right)^2-2](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D-%5Cleft%28x%2B1%5Cright%29%5E2-2)
Openness
- It OPENS DOWN, as 'a=-1' is negative.
Vertex
![\mathrm{Rewrite}\:y=-\left(x+1\right)^2-2\:\mathrm{in\:the\:form}\:y=ax^2+bx+c](https://tex.z-dn.net/?f=%5Cmathrm%7BRewrite%7D%5C%3Ay%3D-%5Cleft%28x%2B1%5Cright%29%5E2-2%5C%3A%5Cmathrm%7Bin%5C%3Athe%5C%3Aform%7D%5C%3Ay%3Dax%5E2%2Bbx%2Bc)
![y=-x^2-2x-3](https://tex.z-dn.net/?f=y%3D-x%5E2-2x-3)
![a=-1,\:b=-2,\:c=-3](https://tex.z-dn.net/?f=a%3D-1%2C%5C%3Ab%3D-2%2C%5C%3Ac%3D-3)
![x_v=-\frac{\left(-2\right)}{2\left(-1\right)}](https://tex.z-dn.net/?f=x_v%3D-%5Cfrac%7B%5Cleft%28-2%5Cright%29%7D%7B2%5Cleft%28-1%5Cright%29%7D)
![x_v=-1](https://tex.z-dn.net/?f=x_v%3D-1)
![\mathrm{Plug\:in}\:\:x_v=-1\:\mathrm{to\:find\:the}\:y_v\:\mathrm{value}](https://tex.z-dn.net/?f=%5Cmathrm%7BPlug%5C%3Ain%7D%5C%3A%5C%3Ax_v%3D-1%5C%3A%5Cmathrm%7Bto%5C%3Afind%5C%3Athe%7D%5C%3Ay_v%5C%3A%5Cmathrm%7Bvalue%7D)
![y_v=-2](https://tex.z-dn.net/?f=y_v%3D-2)
So vertex is:
![\left(-1,\:-2\right)](https://tex.z-dn.net/?f=%5Cleft%28-1%2C%5C%3A-2%5Cright%29)
Horizontal Translation
moves the graph LEFT 1 unit.
Vertical Translation
moves the graph DOWN 2 unit.
Stretch or Compress Vertically
As
< 0, so it is either stretched or compressed.
Q # 3
Explanation:
![f\left(x\right)=\frac{1}{3}\left(x-4\right)^2+6](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D%5Cfrac%7B1%7D%7B3%7D%5Cleft%28x-4%5Cright%29%5E2%2B6)
It OPENS UP, as 'a=1/3' is positive.
Vertex
![\mathrm{Rewrite}\:y=\frac{1}{3}\left(x-4\right)^2+6\:\mathrm{in\:the\:form}\:y=ax^2+bx+c](https://tex.z-dn.net/?f=%5Cmathrm%7BRewrite%7D%5C%3Ay%3D%5Cfrac%7B1%7D%7B3%7D%5Cleft%28x-4%5Cright%29%5E2%2B6%5C%3A%5Cmathrm%7Bin%5C%3Athe%5C%3Aform%7D%5C%3Ay%3Dax%5E2%2Bbx%2Bc)
![y=\frac{1\cdot \:x^2}{3}-\frac{8x}{3}+\frac{34}{3}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B1%5Ccdot%20%5C%3Ax%5E2%7D%7B3%7D-%5Cfrac%7B8x%7D%7B3%7D%2B%5Cfrac%7B34%7D%7B3%7D)
![a=\frac{1}{3},\:b=-\frac{8}{3},\:c=\frac{34}{3}](https://tex.z-dn.net/?f=a%3D%5Cfrac%7B1%7D%7B3%7D%2C%5C%3Ab%3D-%5Cfrac%7B8%7D%7B3%7D%2C%5C%3Ac%3D%5Cfrac%7B34%7D%7B3%7D)
![x_v=-\frac{\left(-\frac{8}{3}\right)}{2\left(\frac{1}{3}\right)}](https://tex.z-dn.net/?f=x_v%3D-%5Cfrac%7B%5Cleft%28-%5Cfrac%7B8%7D%7B3%7D%5Cright%29%7D%7B2%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%7D)
Finding ![y_v](https://tex.z-dn.net/?f=y_v)
![y_v=\frac{1\cdot \:4^2}{3}-\frac{8\cdot \:4}{3}+\frac{34}{3}](https://tex.z-dn.net/?f=y_v%3D%5Cfrac%7B1%5Ccdot%20%5C%3A4%5E2%7D%7B3%7D-%5Cfrac%7B8%5Ccdot%20%5C%3A4%7D%7B3%7D%2B%5Cfrac%7B34%7D%7B3%7D)
So vertex is:
![\left(4,\:6\right)](https://tex.z-dn.net/?f=%5Cleft%284%2C%5C%3A6%5Cright%29)
Horizontal Translation
moves the graph RIGHT 4 units.
Vertical Translation
moves the graph UP 6 unit.
Stretch or Compress Vertically
As
, so it the graph is vertically compressed by a factor of 1/3.
Check the attached comparison graphs.
Q # 4
Explanation:
Given the function
![f\left(x\right)=-\left(x+3\right)^2](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D-%5Cleft%28x%2B3%5Cright%29%5E2)
It OPENS DOWN, as 'a=-1' is negative.
Vertex
The vertex of an up-down facing parabola of the form ![y=a\left(x-m\right)\left(x-n\right)](https://tex.z-dn.net/?f=y%3Da%5Cleft%28x-m%5Cright%29%5Cleft%28x-n%5Cright%29)
is the average of the zeros ![x_v=\frac{m+n}{2}](https://tex.z-dn.net/?f=x_v%3D%5Cfrac%7Bm%2Bn%7D%7B2%7D)
![y=-\left(x+3\right)^2](https://tex.z-dn.net/?f=y%3D-%5Cleft%28x%2B3%5Cright%29%5E2)
![a=-1,\:m=-3,\:n=-3](https://tex.z-dn.net/?f=a%3D-1%2C%5C%3Am%3D-3%2C%5C%3An%3D-3)
![x_v=\frac{m+n}{2}](https://tex.z-dn.net/?f=x_v%3D%5Cfrac%7Bm%2Bn%7D%7B2%7D)
![x_v=\frac{\left(-3\right)+\left(-3\right)}{2}](https://tex.z-dn.net/?f=x_v%3D%5Cfrac%7B%5Cleft%28-3%5Cright%29%2B%5Cleft%28-3%5Cright%29%7D%7B2%7D)
![x_v=-3](https://tex.z-dn.net/?f=x_v%3D-3)
Finding ![y_v](https://tex.z-dn.net/?f=y_v)
![y_v=-\left(-3+3\right)^2](https://tex.z-dn.net/?f=y_v%3D-%5Cleft%28-3%2B3%5Cright%29%5E2)
![y_v=0](https://tex.z-dn.net/?f=y_v%3D0)
So vertex is:
![\left(-3,\:0\right)](https://tex.z-dn.net/?f=%5Cleft%28-3%2C%5C%3A0%5Cright%29)
Horizontal Translation
moves the graph LEFT 3 units.
Vertical Translation
does not move the graph vertically.
Stretch or Compress Vertically
As
, so it the graph is either vertically stretched or compressed.
Q # 5
Explanation:
![f\left(x\right)=\left(x+5\right)^2-3](https://tex.z-dn.net/?f=f%5Cleft%28x%5Cright%29%3D%5Cleft%28x%2B5%5Cright%29%5E2-3)
Openness
- It OPENS UP, as 'a=1' is positive.
Vertex
![\mathrm{Rewrite}\:y=\left(x+5\right)^2-3\:\mathrm{in\:the\:form}\:y=ax^2+bx+c](https://tex.z-dn.net/?f=%5Cmathrm%7BRewrite%7D%5C%3Ay%3D%5Cleft%28x%2B5%5Cright%29%5E2-3%5C%3A%5Cmathrm%7Bin%5C%3Athe%5C%3Aform%7D%5C%3Ay%3Dax%5E2%2Bbx%2Bc)
![y=x^2+10x+22](https://tex.z-dn.net/?f=y%3Dx%5E2%2B10x%2B22)
![a=1,\:b=10,\:c=22](https://tex.z-dn.net/?f=a%3D1%2C%5C%3Ab%3D10%2C%5C%3Ac%3D22)
![x_v=-\frac{10}{2\cdot \:1}](https://tex.z-dn.net/?f=x_v%3D-%5Cfrac%7B10%7D%7B2%5Ccdot%20%5C%3A1%7D)
![x_v=-5](https://tex.z-dn.net/?f=x_v%3D-5)
Finding ![y_v](https://tex.z-dn.net/?f=y_v)
![y_v=\left(-5\right)^2+10\left(-5\right)+22](https://tex.z-dn.net/?f=y_v%3D%5Cleft%28-5%5Cright%29%5E2%2B10%5Cleft%28-5%5Cright%29%2B22)
So vertex is:
![\left(-5,\:-3\right)](https://tex.z-dn.net/?f=%5Cleft%28-5%2C%5C%3A-3%5Cright%29)
Horizontal Translation
moves the graph LEFT 5 units.
Vertical Translation
moves the graph DOWN 3 unit.
Stretch or Compress Vertically
As
, so it does not affect the stretchiness or compression.
Check the attached comparison graphs.
Q # 6
THE DETAILS OF COMPLETE SOLUTION OF QUESTION 6 IS ATTACHED IN THE DIAGRAM AS THE 5000 CHARACTERS WERE ALREADY FILLED. SO, I solved via the attached figure.
SO, PLEASE CHECK THE LAST FIGURE TO FIND THE COMPLETE SOLUTION OF THE Q#6.