7% of x = 450
0.07×x=450
x=450/0.07
x=6428.6 to the nearest tenth (one decimal place)
Answer: ![\bold{1.\quad y=-\dfrac{1}{2}(x-9)^2+10}](https://tex.z-dn.net/?f=%5Cbold%7B1.%5Cquad%20y%3D-%5Cdfrac%7B1%7D%7B2%7D%28x-9%29%5E2%2B10%7D)
2. x = (y - 1)² + 4
![\bold{3.\quad x=\dfrac{1}{2}(y-2)^2+7}](https://tex.z-dn.net/?f=%5Cbold%7B3.%5Cquad%20x%3D%5Cdfrac%7B1%7D%7B2%7D%28y-2%29%5E2%2B7%7D)
<u>Step-by-step explanation:</u>
Notes: The vertex form of a parabola is y = a(x - h)² + k or x = a(y - k)² + h
- (h, k) is the vertex
- p is the distance from the vertex to the focus
![\bullet\quad a=\dfrac{1}{4p}](https://tex.z-dn.net/?f=%5Cbullet%5Cquad%20a%3D%5Cdfrac%7B1%7D%7B4p%7D)
1)
![\text{Vertex}=(9, 10)\qquad \text{Focus}=\bigg(9,\dfrac{19}{2}\bigg)\\\\\text{Given}:(h, k)=(9, 10)\\\\\\p=focus-vertex=\dfrac{19}{2}-\dfrac{20}{2}=\dfrac{-1}{2}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{2})}=\dfrac{1}{-2}=-\dfrac{1}{2}\\](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%289%2C%2010%29%5Cqquad%20%5Ctext%7BFocus%7D%3D%5Cbigg%289%2C%5Cdfrac%7B19%7D%7B2%7D%5Cbigg%29%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%289%2C%2010%29%5C%5C%5C%5C%5C%5Cp%3Dfocus-vertex%3D%5Cdfrac%7B19%7D%7B2%7D-%5Cdfrac%7B20%7D%7B2%7D%3D%5Cdfrac%7B-1%7D%7B2%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B-1%7D%7B2%7D%29%7D%3D%5Cdfrac%7B1%7D%7B-2%7D%3D-%5Cdfrac%7B1%7D%7B2%7D%5C%5C)
Now input a = -1/2 and (h, k) = (9, 10) into the equation y = a(x - h)² + k
![\bold{y=-\dfrac{1}{2}(x-9)^2+10}](https://tex.z-dn.net/?f=%5Cbold%7By%3D-%5Cdfrac%7B1%7D%7B2%7D%28x-9%29%5E2%2B10%7D)
*****************************************************************************************
2)
![\text{Vertex}=(4, 1)\qquad \text{Focus}=\bigg(\dfrac{17}{4},1\bigg)\\\\\text{Given}:(h, k)=(4, 1)\\\\\\p=focus-vertex=\dfrac{17}{4}-\dfrac{16}{4}=\dfrac{1}{4}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{4})}=\dfrac{1}{1}=1\\](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%284%2C%201%29%5Cqquad%20%5Ctext%7BFocus%7D%3D%5Cbigg%28%5Cdfrac%7B17%7D%7B4%7D%2C1%5Cbigg%29%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%284%2C%201%29%5C%5C%5C%5C%5C%5Cp%3Dfocus-vertex%3D%5Cdfrac%7B17%7D%7B4%7D-%5Cdfrac%7B16%7D%7B4%7D%3D%5Cdfrac%7B1%7D%7B4%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B1%7D%7B4%7D%29%7D%3D%5Cdfrac%7B1%7D%7B1%7D%3D1%5C%5C)
Now input a = 1 and (h, k) = (4, 1) into the equation x = a(y - k)² + h
x = 1(y - 1)² + 4 → x = (y - 1)² + 4
*****************************************************************************************
3)
![\text{Vertex}=(7, 2)\qquad \text{Focus}=\bigg(\dfrac{15}{2},2\bigg)\\\\\text{Given}:(h, k)=(7, 2)\\\\\\p=focus-vertex=\dfrac{15}{2}-\dfrac{14}{2}=\dfrac{1}{2}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{2})}=\dfrac{1}{2}](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%287%2C%202%29%5Cqquad%20%5Ctext%7BFocus%7D%3D%5Cbigg%28%5Cdfrac%7B15%7D%7B2%7D%2C2%5Cbigg%29%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%287%2C%202%29%5C%5C%5C%5C%5C%5Cp%3Dfocus-vertex%3D%5Cdfrac%7B15%7D%7B2%7D-%5Cdfrac%7B14%7D%7B2%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B1%7D%7B2%7D%29%7D%3D%5Cdfrac%7B1%7D%7B2%7D)
Now input a = 1/2 and (h, k) = (7, 2) into the equation x = a(y - k)² + h
![\bold{x=\dfrac{1}{2}(y-2)^2+7}](https://tex.z-dn.net/?f=%5Cbold%7Bx%3D%5Cdfrac%7B1%7D%7B2%7D%28y-2%29%5E2%2B7%7D)
Answer:
The value of k is -4 and the equation of the line is
.
Step-by-step explanation:
The equation of the line is defined by the following formula:
(1)
Where:
- Independent variable.
- Dependent variable.
- y-Intercept.
- Slope.
In addition, the slope of the equation of the line (
) can be known from two distinct points:
(2)
Where
and
are the coordinates of the reference point.
By applying (2) in (1), we derive the resulting expression:
![y = \frac{y-y_{o}}{x-x_{o}}\cdot x + b](https://tex.z-dn.net/?f=y%20%3D%20%5Cfrac%7By-y_%7Bo%7D%7D%7Bx-x_%7Bo%7D%7D%5Ccdot%20x%20%2B%20b)
![y\cdot (x-x_{o}) = (y-y_{o})\cdot x + b\cdot (x-x_{o})](https://tex.z-dn.net/?f=y%5Ccdot%20%28x-x_%7Bo%7D%29%20%3D%20%28y-y_%7Bo%7D%29%5Ccdot%20x%20%2B%20b%5Ccdot%20%28x-x_%7Bo%7D%29)
![x\cdot y -y\cdot x_{o} = x\cdot y -x\cdot y_{o}+b\cdot x -b\cdot x_{o}](https://tex.z-dn.net/?f=x%5Ccdot%20y%20-y%5Ccdot%20x_%7Bo%7D%20%3D%20x%5Ccdot%20y%20-x%5Ccdot%20y_%7Bo%7D%2Bb%5Ccdot%20x%20-b%5Ccdot%20x_%7Bo%7D)
![(b-y)\cdot x_{o} = b\cdot x -x\cdot y_{o}](https://tex.z-dn.net/?f=%28b-y%29%5Ccdot%20x_%7Bo%7D%20%3D%20b%5Ccdot%20x%20-x%5Ccdot%20y_%7Bo%7D)
(3)
If we know that
,
,
,
and
, then the value of
is:
![(10-9)\cdot (k+10)+2\cdot (-2\cdot k -1) = (10)\cdot (2)](https://tex.z-dn.net/?f=%2810-9%29%5Ccdot%20%28k%2B10%29%2B2%5Ccdot%20%28-2%5Ccdot%20k%20-1%29%20%3D%20%2810%29%5Ccdot%20%282%29)
![k+10 -4\cdot k -2 = 20](https://tex.z-dn.net/?f=k%2B10%20-4%5Ccdot%20k%20-2%20%3D%2020)
![-3\cdot k +8 = 20](https://tex.z-dn.net/?f=-3%5Ccdot%20k%20%2B8%20%3D%2020)
![-3\cdot k = 12](https://tex.z-dn.net/?f=-3%5Ccdot%20k%20%3D%2012)
![k = -4](https://tex.z-dn.net/?f=k%20%3D%20-4)
The value of the slope is:
![m = \frac{9-7}{2-6}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B9-7%7D%7B2-6%7D)
![m = -\frac{1}{2}](https://tex.z-dn.net/?f=m%20%3D%20-%5Cfrac%7B1%7D%7B2%7D)
The equation of the line is
.
<em>Complete Question:</em>
<em>Alicia is writing the program for a video game. For one part of the game she uses the rule (x,y)->(x-3,y+4) to move points on the screen. </em>
<em>A) what output does the rule give when the input is (-6,0)? </em>
<em>B)What output does the rule give when the input is (3,-4)? </em>
<em>C) Is the rule a function? Explain why it is or why it is not
</em>
Answer:
a. The output is (-9,4)
b. The output is (0,0)
c. See Explanation
Step-by-step explanation:
Given
Rule: (x,y)->(x - 3, y + 4)
Solving (a):
Inputs:
![x = -6](https://tex.z-dn.net/?f=x%20%3D%20-6)
![y = 0](https://tex.z-dn.net/?f=y%20%3D%200)
The outputs is as follows;
![x => x - 3](https://tex.z-dn.net/?f=x%20%3D%3E%20x%20-%203)
![x => -6 - 3](https://tex.z-dn.net/?f=x%20%3D%3E%20-6%20-%203)
![x => -9](https://tex.z-dn.net/?f=x%20%3D%3E%20-9)
![y=> y + 4](https://tex.z-dn.net/?f=y%3D%3E%20y%20%2B%204)
![y=> 0 + 4](https://tex.z-dn.net/?f=y%3D%3E%200%20%2B%204)
![y=> 4](https://tex.z-dn.net/?f=y%3D%3E%204)
Hence:
The output is (-9,4)
Solving (b):
Inputs:
![x = 3](https://tex.z-dn.net/?f=x%20%3D%203)
![y = -4](https://tex.z-dn.net/?f=y%20%3D%20-4)
The outputs is as follows;
![x => x - 3](https://tex.z-dn.net/?f=x%20%3D%3E%20x%20-%203)
![x=>3-3](https://tex.z-dn.net/?f=x%3D%3E3-3)
![x = 0](https://tex.z-dn.net/?f=x%20%3D%200)
![y=> y + 4](https://tex.z-dn.net/?f=y%3D%3E%20y%20%2B%204)
![y=>-4+4](https://tex.z-dn.net/?f=y%3D%3E-4%2B4)
![y=>0](https://tex.z-dn.net/?f=y%3D%3E0)
Hence:
The output is (0,0)
Solving (c):
The function is a rule and the rule is that:
It shifts the graph left by 3 units and up by 4 units
The slope/rate of change is m=2