write a rule for g of f(x)

Question

write a rule for g that represents a translation 3 units right and 1 unit up, followed by a vertical stretch by a factor of 8 of the graph f(x)= lnx

Answer 1

g(x) = a·ln(b(x-c)) + d

1. a = vertical stretch; reflects across x-axis if a < 0 = 8
2. b = horizontal stretch by factor 1/b; reflects across y-axis if b < 0 = none = 1
3. c = horizontal translation = 3
4. d = vertical translation = 1

g(x) = 8·ln(x-3) + 1

Answer 2

To translate or scale a plot, replace the appropriate variable with an expression that requires the variable to be modified to achieve the stated result.

example: y = x. To shift this to the right by 3 units, replace x with x - 3.

y = x - 3. x now has to be 3 units larger to satisfy the equation.

To stretch, replace the appropriate variable with one DIVIDED by the stretch factor.

example: y - 3 = x. To stretch this vertically by a factor of 3, replace y with y/3, Now y must be larger to satisfy the equation.

Note that y/3 - 3 = x is the same as y = 3*( x + 3 ).

For the stated problem:

y = f(x) = ln(x)

Shift 3 units right and 1 unit up:

y - 1 = ln (x - 3)

Scale up (stretch) by a factor of 8 vertically:

y/8 - 1 = ln (x - 3)

Note that this would normally be written as:

y = 8*( ln (x - 3) + 1 )