Transform the graph of f(x)=3^x to sketch g(x)=3^−(x+1) −2. Show table of values and each transformation clearly to get full marks. (15,marks)?

well, ya got a table?

Maybe these graphs will help you out in determining the transformations required.

http://www.wolframalpha.com/input/?i=plot+y%3D3%5Ex%2C+y%3D3%5E%28%E2%88%92%28x%2B1%29%29+%E2%88%922

To transform the graph of the function f(x) = 3^x to sketch g(x) = 3^-(x+1) - 2, we will follow a step-by-step process that involves applying each transformation to the original graph.

Step 1: Start with the graph of f(x) = 3^x
Let's first create a table of values for f(x), which will help us plot points on the graph:

x | f(x)
----------
-2 | 1/9
-1 | 1/3
0 | 1
1 | 3
2 | 9

Plotting these points on a Cartesian plane will give us the graph of f(x) = 3^x. It is an exponential function that passes through the points (-2, 1/9), (-1, 1/3), (0, 1), (1, 3), and (2, 9).

Step 2: Apply the transformation g(x) = 3^-(x+1) - 2
Now, let's apply the transformation to the graph of f(x) by following these steps:

a) Reflection over the x-axis: To reflect the graph over the x-axis, we need to negate the y-values of the original points. So, the new y-values will be -(original y-values):

x | f(x) | -f(x)
-------------------
-2 | 1/9 | -1/9
-1 | 1/3 | -1/3
0 | 1 | -1
1 | 3 | -3
2 | 9 | -9

Plot the points (-2, -1/9), (-1, -1/3), (0, -1), (1, -3), and (2, -9) on the same Cartesian plane as the graph of f(x) = 3^x.

b) Translation to the left by 1 unit: To perform this translation, we need to subtract 1 from the x-values of the points we obtained in the previous step. So, the new x-values will be (original x-values - 1):

x | f(x) | -f(x) | x+1
----------------------
-2 | 1/9 | -1/9 | -1
-1 | 1/3 | -1/3 | 0
0 | 1 | -1 | 1
1 | 3 | -3 | 2
2 | 9 | -9 | 3

Plot the new points (-1, -1/9), (0, -1/3), (1, -1), (2, -3), and (3, -9) on the graph.

c) Vertical shift downward by 2 units: This transformation involves subtracting 2 from the y-values of the points we plotted in the previous step. So, the new y-values will be (original y-values - 2):

x | g(x)
----------------
-1 | -1/9 - 2
0 | -1/3 - 2
1 | -1 - 2
2 | -3 - 2
3 | -9 - 2

Simplifying these expressions, we get:

x | g(x)
----------------
-1 | -19/9
0 | -7/3
1 | -3
2 | -5
3 | -11

Plot these new points (-1, -19/9), (0, -7/3), (1, -3), (2, -5), and (3, -11) on the graph.

Connecting these points will give us the sketch of the transformed function g(x) = 3^-(x+1) - 2.

Remember to label the axes, indicate the transformations performed, and provide a clear scale on the graph to ensure full marks.