how can you use transformations to graph this function?\n( y = 3 cdot 7 ^ { - x } + 2 )\nexplain your steps.

Explanation:

Step1: Identify the parent function

The parent function is (y = 7^{x}).

Step2: Reflect over the y - axis

For (y = 7^{-x}), we use the transformation (y = f(-x)) which reflects the graph of (y = f(x)) (here (f(x)=7^{x})) over the (y) - axis.

Step3: Vertical stretch

For (y = 3\cdot7^{-x}), we use the transformation (y = a\cdot f(x)) ((a = 3>1)) which vertically stretches the graph of (y = 7^{-x}) by a factor of (3).

Step4: Vertical shift

For (y=3\cdot7^{-x}+2), we use the transformation (y = f(x)+k) ((k = 2)) which shifts the graph of (y = 3\cdot7^{-x}) up by (2) units.

Answer:

Start with the parent function (y = 7^{x}). Reflect it over the (y) - axis to get (y = 7^{-x}). Vertically stretch the resulting graph by a factor of (3) to obtain (y = 3\cdot7^{-x}). Finally, shift the graph of (y = 3\cdot7^{-x}) up by (2) units to graph (y = 3\cdot7^{-x}+2).