The graph y=-2\left(\frac{3}{2}-e^{3-x}\right) by: a) Performing the necessary algebra so that

Annette Arroyo

Annette Arroyo

Answered question

2021-08-10

The graph y=2(32e3x) by:
a) Performing the necessary algebra so that the function is in the proper form (i.e., the transformations are in the proper order).
b) Listing the transformations in the order that they are to be applied.
c) Marking the key point and horizontal asymptote.

Answer & Explanation

Brighton

Brighton

Skilled2021-08-11Added 103 answers

a) Rewrite the given function as
f(x)=3+2×e(x3)
Now, f(x) is in the required form. b) f(x)=3+2×e(x3),
f(x) is obtained from the basic exponential function, g(x)=ex, by the following sequence of transformations, in the given order:
1) shifting along x-axis: xx3
2) Scaling the y-coordinate: y2y
3) Translation vertically (below) by 2 units: yy2
c) f(x)=3+2×e(x3),
Key points:
1. limxf(x)=.
2. limxf(x)=3. (horizontal asymptote)
3. When x=0, y=3+2e3
4. y=0, when e3x=1.53x=ln(1.5)
x=3ln(1.5)
5.y=2×e3x<0x. Thus y=f(x)
is monotonically decreasing on (,)
from to 3 (asymptotic limit)
So, no vertical asymptote and the horizontal asymptote is y=3

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