Answer: 1. y = 2(x + 4)² - 3
![\bold{2.\quad y=-\dfrac{1}{3}(x+8)^2-7}](https://tex.z-dn.net/?f=%5Cbold%7B2.%5Cquad%20y%3D-%5Cdfrac%7B1%7D%7B3%7D%28x%2B8%29%5E2-7%7D)
![\bold{3.\quad y=-\dfrac{1}{2}(x-7)^2+1}](https://tex.z-dn.net/?f=%5Cbold%7B3.%5Cquad%20y%3D-%5Cdfrac%7B1%7D%7B2%7D%28x-7%29%5E2%2B1%7D)
<u>Step-by-step explanation:</u>
Notes: The vertex form of a parabola is y = a(x - h)² + k
- (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}=(-4,-3)\qquad \text{Directrix}:y=-\dfrac{25}{8}\\\\\text{Given}:(h, k)=(-4, 3)\\\\p=\dfrac{-24}{8}-\dfrac{-25}{8}=\dfrac{1}{8}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{8})}=\dfrac{1}{\frac{1}{2}}=2](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%28-4%2C-3%29%5Cqquad%20%5Ctext%7BDirectrix%7D%3Ay%3D-%5Cdfrac%7B25%7D%7B8%7D%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%28-4%2C%203%29%5C%5C%5C%5Cp%3D%5Cdfrac%7B-24%7D%7B8%7D-%5Cdfrac%7B-25%7D%7B8%7D%3D%5Cdfrac%7B1%7D%7B8%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B1%7D%7B8%7D%29%7D%3D%5Cdfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D2)
Now input a = 2 and (h, k) = (-4, -3) into the equation y = a(x - h)² + k
y = 2(x + 4)² - 3
******************************************************************************************
2)
![\text{Vertex}=(-8,-7)\qquad \text{Directrix}:y=-\dfrac{-25}{4}\\\\\text{Given}:(h, k)=(-8, -7)\\\\p=\dfrac{-28}{4}-\dfrac{-25}{4}=\dfrac{-3}{4}}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-3}{4})}=\dfrac{1}{-3}=-\dfrac{1}{3}](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%28-8%2C-7%29%5Cqquad%20%5Ctext%7BDirectrix%7D%3Ay%3D-%5Cdfrac%7B-25%7D%7B4%7D%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%28-8%2C%20-7%29%5C%5C%5C%5Cp%3D%5Cdfrac%7B-28%7D%7B4%7D-%5Cdfrac%7B-25%7D%7B4%7D%3D%5Cdfrac%7B-3%7D%7B4%7D%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B-3%7D%7B4%7D%29%7D%3D%5Cdfrac%7B1%7D%7B-3%7D%3D-%5Cdfrac%7B1%7D%7B3%7D)
Now input a = -1/3 and (h, k) = (-8, -7) into the equation y = a(x - h)² + k
![\bold{y=-\dfrac{1}{3}(x+8)^2-7}](https://tex.z-dn.net/?f=%5Cbold%7By%3D-%5Cdfrac%7B1%7D%7B3%7D%28x%2B8%29%5E2-7%7D)
******************************************************************************************
3)
![\text{Focus}=\bigg(7,\dfrac{1}{2}\bigg)\qquad \text{Directrix}:y=\dfrac{3}{2}](https://tex.z-dn.net/?f=%5Ctext%7BFocus%7D%3D%5Cbigg%287%2C%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%29%5Cqquad%20%5Ctext%7BDirectrix%7D%3Ay%3D%5Cdfrac%7B3%7D%7B2%7D)
The midpoint of the focus and directrix is the y-coordinate of the vertex:
![\dfrac{focus+directrix}{2}=\dfrac{\frac{1}{2}+\frac{3}{2}}{2}=\dfrac{\frac{4}{2}}{2}=\dfrac{2}{2}=1](https://tex.z-dn.net/?f=%5Cdfrac%7Bfocus%2Bdirectrix%7D%7B2%7D%3D%5Cdfrac%7B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B3%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B%5Cfrac%7B4%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B2%7D%7B2%7D%3D1)
The x-coordinate of the vertex is given in the focus as 7
(h, k) = (7, 1)
Now let's find the a-value:
![p=\dfrac{2}{2}-\dfrac{3}{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=p%3D%5Cdfrac%7B2%7D%7B2%7D-%5Cdfrac%7B3%7D%7B2%7D%3D%5Cdfrac%7B-1%7D%7B2%7D%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)
Now input a = -1/2 and (h, k) = (7, 1) into the equation y = a(x - h)² + k
![\bold{y=-\dfrac{1}{2}(x-7)^2+1}](https://tex.z-dn.net/?f=%5Cbold%7By%3D-%5Cdfrac%7B1%7D%7B2%7D%28x-7%29%5E2%2B1%7D)
Let's solve for x.
Step 1: Add -3x to both sides.
Step 2: Add -8 to both sides.
Step 3: Divide both sides by 4.
Use long division to find the quotient below (x²+15x+56)÷(x+8) = x + 7
The equations will be y-b/ x =m/1, y-b=m(x) and y=mx +b
Step-by-step explanation:
In the diagram you can show the two triangles are similar by showing that the ratio of rise to rum is equal.This is
y-b/ x =m/1 Multiplying both sides by x y-b=m(x) Adding b on both sides will eliminate the b on the left side as; y-b+b= mx+b y=mx +b where m is the slope and b is the y-intercept
Learn More
To write the slope-intercept equation : brainly.com/question/10676927
The slope intercept equation : brainly.com/question/11514369
Keywords : slope intercept equation
#LearnwithBrainly
Interval notation uses an ordered pair inside brackets. The ordered pair is ...
... lower end of the interval, upper end of the interval
The brackets are square if the endpoint is included (solid dot, or "≤"), and they are curved (parentheses) if the endpoint is not included (open dot, or "<"). The brackets don't have to match.
Here, your interval is -4 ≤ x ≤ -1, so we use square brackets around the end values -4 and -1.
... [-4, -1]