Answer:
The graph is attached; it is the graph of f(x) = x.
Step-by-step explanation:
Two functions are inverses if their composites form the function f(x) = x.
Take, for example, the function f(x) = 2x-4. We can write this as y=2x-4.
To find the inverse, we isolate x. To do this, first add 4 to each side:
y+4 = 2x-4+4
y+4 = 2x
Divide both sides by 2:
(y+4)/2 = 2x/2
y/2 + 4/2 = x
1/2y + 2 = x
Swap x and y, and the inverse is
y=1/2x+2
This can be written as g(x).
The composite of these two functions, f(g(x)), is:
f(1/2x+2) = 2(1/2x+2)-4
= 2(1/2x)+2(2)-4
=1x+4-4
= x
The composite of two functions is always f(x) = x.
This means the graph will have a slope of 1 and a y-intercept of 0.
Right angle/90° angle, i believe
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)
Answer:
x = -9
Step-by-step explanation:
12 + 2x + 24 = 27 + x
(12 + 24) + 2x = 27 + x
36 + 2x = 27 + x
subtract 27 from both sides
36-27 + 2x = 27-27 + x
9 + 2x = x
subtract x from both sides
9 + x = 0
subtract 9 from both sides
x = -9
(the steps could've been faster but that shows the most work)
check:
12 + 2(-9) + 24 = 27 + (-9)
36 - 18 = 18
18 = 18
True!
so -9 is correct