suppose that f and g are functions. Suppose further that we can write g in terms of f as follows:
g(x)=f(x-3)+5
List a sequence of transformations that will transform the graph of f into a graph of g.
I am at a loss of how to even break this down into a description with words. Can anyone explain this to me? Thank you, thank you in advance.
We begin by starting with f(x) and building up the new expressing by
following this order of operations. I am giving you all of them but
for this problem you only need to use steps 5 and 6:
1. Replace x by -x to reflect the graph horizontally across the y-axis.
2. Replace x by ax, a > 0 to shrink the graph horizontally if a > 1 or stretch
the graph horizontally if positive number a < 1, by a factor of a.
3. Multiply the entire expression by positive number b to shrink the graph
vertically if b < 1, or stretch the graph vertically if b > 1, by a factor of
b.
4. Multiply the entire expression by -1 to reflect the graph across the x-axis.
5. Replace x by (x-c), c > 0, to shift the graph right c units, or by (x+c)
to shift the graph left c units.
6. Add d, d > 0, to the right side of the entire expression to shift the graph
upward by d units, or subtract d from the right side to shift the graph
downward by d units.
As I said above, you only need steps 5 and 6, to go from
f(x)
to
f(x-3)+5
We start with
f(x)
We use (5)
We replace x by (x-3), 3 > 0, to shift the graph right 3 units
We get:
f(x-3)
We use (6)
We add 5, d > 0, to the right side of the entire expression to shift the graph
upward by 5 units.
We get
f(x-3)+5
So the transformations are
Shift the graph right 3 units, then shift the graph upward 5 units.
Edwin