Let f(x)=7^x and g(x)=7^x+2-4. Which transformations are needed to transform f(x) to the graph g(x)?
1 answer:
Answer:
4. vertical translation 4 units down
6. horizontal translation 2 units to the right
Step-by-step explanation:
Given and
1. Adding 2 to x means horizontal translation 2 units to the right.
2. Subtracting 4 from the function means vertical translation 4 units down.
So, to transform the graph of the function f(x) to the graph of the function g(x), you need to translate the graph of the function f(x) horizontally 2 units to the right and vertically 4 units down.
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The distance between a point 
on the given plane and the point (0, 2, 4) is
but since
and 
share critical points, we can instead consider the problem of optimizing subject to 
.
The Lagrangian is
with partial derivatives (set equal to 0)
Solve for
:
which gives the critical point
We can confirm that this is a minimum by checking the Hessian matrix of:
is positive definite (we see its determinant and the determinants of its leading principal minors are positive), which indicates that there is a minimum at this critical point.
At this point, we get a distance from (0, 2, 4) of
but since
The Lagrangian is
with partial derivatives (set equal to 0)
Solve for
which gives the critical point
We can confirm that this is a minimum by checking the Hessian matrix of
At this point, we get a distance from (0, 2, 4) of