The attached picture is a summary of all the six transformations you'd do.
Any change outside the f(x) notation impacts y-values of points on the graph.
Any changes inside the f(x) notation impacts x-values of points on the graph.
The trick is that the inside changes are usually the opposite of what you'd expect to have happen.
7. y=f(x)+8
This is an outside change. You're adding 8 to all the y-values of points on the graph. This will shift your entire graph up 8 units.
8. y=f(x+6)
This is an inside change. Because it says "+6", you want to think, "Ah! That means I'll actually subtract 6 from the x-value of every point on the graph." This graph is shifted 6 units to the left.
9. y=-f(x)
Inside change, impacts y-values. Every y-value will be given the opposite signs. Negatives become positive and positives become negative. This will flip your graph over the x-axis.
10. y = f(-x) + 5
Give this one a shot on your own first in a comment and I'll let you know how you did.
11. y = - 3 f(x-3)
There are three things happening. A negative on the outside, multiplying by 3 on the outside, and subtracting 3 inside. What will each of those do individually? Take a shot on this one and let me know what you think.
12. y = 1/2 f( 1/2 x )
Again, three changes. Try this one and let me know what you think. Remember multiplying by 1/2 inside really means you'll do the opposite of multiplying by 1/2.
Answer:
x = 12.1
Step-by-step explanation:
20 − (z + 7) = 0.9
Distribute the minus sign
20 -x -7 = .9
Combine like terms
13 -x = .9
Subtract 13 from each side
-x = -12.1
x = 12.1
Answer:
Metaphor
Step-by-step explanation:
This is a comparison between the ocean and a magnet, but it did not use like or as, so it is a metaphor.
Answer:
![c^2 = 9dp](https://tex.z-dn.net/?f=c%5E2%20%3D%209dp)
Step-by-step explanation:
Given
![dx^2 + cx + p = 0](https://tex.z-dn.net/?f=dx%5E2%20%2B%20cx%20%2B%20p%20%3D%200)
Let the roots be
and ![\beta](https://tex.z-dn.net/?f=%5Cbeta)
So:
![\alpha = 2\beta](https://tex.z-dn.net/?f=%5Calpha%20%3D%202%5Cbeta)
Required
Determine the relationship between d, c and p
![dx^2 + cx + p = 0](https://tex.z-dn.net/?f=dx%5E2%20%2B%20cx%20%2B%20p%20%3D%200)
Divide through by d
![\frac{dx^2}{d} + \frac{cx}{d} + \frac{p}{d} = 0](https://tex.z-dn.net/?f=%5Cfrac%7Bdx%5E2%7D%7Bd%7D%20%2B%20%5Cfrac%7Bcx%7D%7Bd%7D%20%2B%20%5Cfrac%7Bp%7D%7Bd%7D%20%3D%200)
![x^2 + \frac{c}{d}x + \frac{p}{d} = 0](https://tex.z-dn.net/?f=x%5E2%20%2B%20%5Cfrac%7Bc%7D%7Bd%7Dx%20%2B%20%5Cfrac%7Bp%7D%7Bd%7D%20%3D%200)
A quadratic equation has the form:
![x^2 - (\alpha + \beta)x + \alpha \beta = 0](https://tex.z-dn.net/?f=x%5E2%20-%20%28%5Calpha%20%2B%20%5Cbeta%29x%20%2B%20%5Calpha%20%5Cbeta%20%3D%200)
So:
![x^2 - (2\beta+ \beta)x + \beta*\beta = 0](https://tex.z-dn.net/?f=x%5E2%20-%20%282%5Cbeta%2B%20%5Cbeta%29x%20%2B%20%5Cbeta%2A%5Cbeta%20%3D%200)
![x^2 - (3\beta)x + \beta^2 = 0](https://tex.z-dn.net/?f=x%5E2%20-%20%283%5Cbeta%29x%20%2B%20%5Cbeta%5E2%20%3D%200)
So, we have:
-- (1)
and
-- (2)
Make
the subject in (1)
![\frac{c}{d} = -3\beta](https://tex.z-dn.net/?f=%5Cfrac%7Bc%7D%7Bd%7D%20%3D%20-3%5Cbeta)
![\beta = -\frac{c}{3d}](https://tex.z-dn.net/?f=%5Cbeta%20%3D%20-%5Cfrac%7Bc%7D%7B3d%7D)
Substitute
in (2)
![\frac{p}{d} = (-\frac{c}{3d})^2](https://tex.z-dn.net/?f=%5Cfrac%7Bp%7D%7Bd%7D%20%3D%20%28-%5Cfrac%7Bc%7D%7B3d%7D%29%5E2)
![\frac{p}{d} = \frac{c^2}{9d^2}](https://tex.z-dn.net/?f=%5Cfrac%7Bp%7D%7Bd%7D%20%3D%20%5Cfrac%7Bc%5E2%7D%7B9d%5E2%7D)
Multiply both sides by d
![d * \frac{p}{d} = \frac{c^2}{9d^2}*d](https://tex.z-dn.net/?f=d%20%2A%20%5Cfrac%7Bp%7D%7Bd%7D%20%3D%20%5Cfrac%7Bc%5E2%7D%7B9d%5E2%7D%2Ad)
![p = \frac{c^2}{9d}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7Bc%5E2%7D%7B9d%7D)
Cross Multiply
![9dp = c^2](https://tex.z-dn.net/?f=9dp%20%3D%20c%5E2)
or
![c^2 = 9dp](https://tex.z-dn.net/?f=c%5E2%20%3D%209dp)
Hence, the relationship between d, c and p is: ![c^2 = 9dp](https://tex.z-dn.net/?f=c%5E2%20%3D%209dp)