sketch the graph of the function y = 3^x+1 - 9. be sure to label all intercepts and asymptotes, and show at…

Explanation:

Step1: Find the y - intercept

Set (x = 0). Then (y=3^{0 + 1}-9=3 - 9=-6). So the y - intercept is ((0,-6)).

Step2: Find the x - intercept

Set (y = 0). Then (3^{x + 1}-9 = 0), which can be rewritten as (3^{x+1}=9=3^{2}). So (x + 1=2), and (x = 1). The x - intercept is ((1,0)).

Step3: Find the horizontal asymptote

As (x\to-\infty), (3^{x + 1}\to0). So (y = 3^{x+1}-9\to - 9). The horizontal asymptote is (y=-9).

Step4: Find an additional point

Let (x=-1). Then (y=3^{-1 + 1}-9=1 - 9=-8). The point is ((-1,-8)).

To sketch the graph: Plot the y - intercept ((0,-6)), the x - intercept ((1,0)), the additional point ((-1,-8)) and draw a curve approaching the horizontal asymptote (y = - 9) as (x\to-\infty).

Answer:

The y - intercept is ((0,-6)), the x - intercept is ((1,0)), the horizontal asymptote is (y=-9) and an additional point is ((-1,-8)). Sketch the graph using these features.