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Maru [420]
3 years ago
12

Write the standard equation for the ellipse with the given conic form.

Mathematics
1 answer:
Andrei [34K]3 years ago
6 0

We complete the square


4 x^2 + y^2 - 16x + 2y  + 13 = 0


4 x^2 - 16x +  y^2 + 2y  = -13


4 (x^2 - 4x) +  (y^2 + 2y)  = -13


4 (x^2 - 4x + 4) +  (y^2 + 2y  +1 )  = -13 +16 + 1


4 (x - 2)^2 +  (y+1)^2 = 4


\dfrac{ (x - 2)^2 }{1^2} + \dfrac{ (y+1)^2}{2^2} = 1


That's standard form with axes 1 and 2



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Describe how to transform the graph of g(x)= ln x into the graph of f(x)= ln (3-x) -2.
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Answer:

The graph of g(x) = ㏑x translated 3 units to the right and then reflected

about the y-axis and then translated 2 units down to form the graph of

f(x) = ㏑(3 - x) - 2

Step-by-step explanation:

* Lets talk about the transformation

- If the function f(x) reflected across the x-axis, then the new

 function g(x) = - f(x)

- If the function f(x) reflected across the y-axis, then the new

 function g(x) = f(-x)

- If the function f(x) translated horizontally to the right  

 by h units, then the new function g(x) = f(x - h)

- If the function f(x) translated horizontally to the left  

 by h units, then the new function g(x) = f(x + h)

- If the function f(x) translated vertically up  

 by k units, then the new function g(x) = f(x) + k

- If the function f(x) translated vertically down  

 by k units, then the new function g(x) = f(x) – k

* lets solve the problem

∵ Graph of g(x) = ㏑x is transformed into graph of f(x) = ㏑(3 - x) - 2

- ㏑x becomes ㏑(3 - x)

∵ ㏑(3 - x) = ㏑(-x + 3)

- Take (-) as a common factor

∴ ㏑(-x + 3) = ㏑[-(x - 3)]

∵ x changed to x - 3

∴ The function g(x) translated 3 units to the right

∵ There is (-) out the bracket (x - 3) that means we change the sign

  of x then we will reflect the function about the y-axis

∴ g(x) translated 3 units to the right and then reflected about the

   y-axis

∵ g(x) changed to f(x) = ㏑(3 - x) - 2

∵ We subtract 2 from g(x) after horizontal translation and reflection

  about y-axis

∴ We translate g(x) 2 units down

∴ g(x) translated 3 units to the right and then reflected about the

   y-axis and then translated 2 units down

* The graph of g(x) = ㏑x translated 3 units to the right and then

  reflected about the y-axis and then translated 2 units down to

  form the graph of f(x) = ㏑(3 - x) - 2

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