SOLUTION: For the equations f(x) = (3)^x and g(x) = -(3)^2x+6-5, what are the transformations applied to f(x) to obtain g(x)?

Algebra ->  Functions -> SOLUTION: For the equations f(x) = (3)^x and g(x) = -(3)^2x+6-5, what are the transformations applied to f(x) to obtain g(x)?       Log On


   

Question 1180811: For the equations f(x) = (3)^x and g(x) = -(3)^2x+6-5, what are the transformations applied to f(x) to obtain g(x)?
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29+=+%283%29%5Ex+
g%28x%29+=+-%283%29%5E%282x%2B6%29-5
g%28x%29+=+-%283%29%5E%282%28x%2B3%29%29-5



the graph is moved left 3 and then reflected over the+x-axis, moved down 5+units

Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


If you are working on a problem like this, your level of knowledge of math should be enough to know that proper use of parentheses is important. The function you show for g(x) is this:

g%28x%29+=+-%283%29%5E2x%2B6-5

It is obvious from the context of the problem that that is not the g(x) that you want. Surely you want this:

g(x) = -(3)^(2x+6)-5: g%28x%29+=+-%283%29%5E%282x%2B6%29-5

To determine the transformations, we need to rewrite the function so that the variable x has coefficient 1:

g%28x%29+=+-%283%29%5E%282%28x%2B3%29%29-5

The transformations are in the order in which the expression would be evaluated for a given value of x.

Parent function: 3^x
graph%28400%2C400%2C-10%2C2%2C-10%2C10%2C3%5Ex%29

(1) Inner parentheses: (x+3)
That shifts the graph 3 units left: 3^(x+3) (green)
graph%28400%2C400%2C-10%2C2%2C-10%2C10%2C3%5Ex%2C3%5E%28x%2B3%29%29

(2) Outer parentheses: 2(x+3)
That compresses (dilates) the graph horizontally by a factor of 2, with the center of the dilation at x=-3: 3^(2(x+3)) (blue)
graph%28400%2C400%2C-10%2C2%2C-10%2C10%2C3%5Ex%2C3%5E%28x%2B3%29%2C3%5E%282%28x%2B3%29%29%29

(3) Multiplication: multiply by -1
That flips the graph about the x-axis: -3^(2(x+3)) (purple)


(4) Addition/subtraction: subtract 5
That moves the graph down 5 units: -3^(2(x+3))-5 (yellow)


Summary....

The transformations, in order, are
shift left 3
compress horizontally by a factor of 2
flip vertically
shift down 5

Answer by ikleyn(52752) About Me  (Show Source):
You can put this solution on YOUR website!
.

Comparing solutions from  @MathLover1  and  @greenestamps,  notice that  @MathLower1

missed one important step  (compressing x-axis).