graphing translated functions\nthe parent function $f(x)=1.5^{x}$ is translated such that the function…

Answer:

The graph of an exponential function of the form (y = a^{x - h}+k) is a transformation of the parent - function (y = a^{x}). For the function (g(x)=1.5^{x + 1}+2), which can be written as (g(x)=1.5^{x-(- 1)}+2), compared to the parent function (f(x)=1.5^{x}):

1. The value of (h=-1) represents a horizontal shift. A negative value of (h) means a shift to the left by 1 unit.
2. The value of (k = 2) represents a vertical shift. A positive value of (k) means a shift up by 2 units.

The parent function (y = 1.5^{x}) has a (y) - intercept at ((0,1)) (since when (x = 0), (y=1.5^{0}=1)).

For the function (g(x)=1.5^{x + 1}+2), when (x=-1), (g(-1)=1.5^{-1 + 1}+2=1 + 2=3). So the (y) - intercept of (g(x)) is at the point ((-1,3))

We can also consider the general shape of an exponential growth function (y = a^{x}) with (a>1) (here (a = 1.5)). The function (g(x)=1.5^{x + 1}+2) is an exponential growth function that has been shifted left 1 unit and up 2 units from the parent function (y = 1.5^{x})

We need to look for a graph that has the general shape of an exponential growth curve ((y) increases as (x) increases) and has a (y) - intercept at ((-1,3))

Explanation:

Step1: Identify the horizontal shift

The function (g(x)=1.5^{x + 1}+2) has (h=-1), so it shifts left 1 unit.

Step2: Identify the vertical shift

The function (g(x)=1.5^{x + 1}+2) has (k = 2), so it shifts up 2 units.

Step3: Find the (y) - intercept of (g(x))

Set (x=-1), (g(-1)=1.5^{-1 + 1}+2=3).

Step4: Analyze the shape of the graph

Since (a = 1.5>1), it is an exponential growth function.