Question

The topic of the assignment is: Graphing Decay Functions I will post a screenshot of the function that I need to graph. The function follows this formula: f(x)=a(b)^(x-h)+k

Answer 1

We are asked to graph the following function:

`y=(1)/(2)((1)/(6))^(x-1)-2`

Before sketching the graph we will simplify the function. To do that we will use the following property of exponentials:

`a^(x+y)=a^xa^y`

Applying the property we get:

`y=(1)/(2)((1)/(6))^x((1)/(6))^(-1)-2`

Now we use the following property:

`((a)/(b))^(-1)=(b)/(a)`

Applying the property we get:

`y=(1)/(2)((1)/(6))^x(6)-2`

Simplifying we get:

`y=3((1)/(6))^x-2`

Now, we will graph the function. To do that we will need to determine three points of the graph.

The first point we will determine it by setting x = 0, to determine the y-intercept:

`\begin{gathered} y=3((1)/(6))^0-2 \\ \\ y=3-2=1 \end{gathered}`

Therefore, the point:

`(x,y)=(0,1)`

is part of the graph. Now, we will set x = 1:

`y=3((1)/(6))^1-2`

Solving the operations:

`y=-(3)/(2)`

Therefore, the point:

`(x,y)=(1,-(3)/(2))`

Now, we will determine the horizontal asymptote. To do that we need to analyze what happens when x goes to infinity. As the values of "x" get larger and larger the value of:

`((1)/(6))^x`

will tend to zero, since would have a large number of products of 1/6 and as the denominator grows the numerator will stay as 1 and therefore, the fraction will go to zero, therefore, the asymptote is:

`y=3(0)-2=-2`

Now, we plot the points and the asymptote, we get the following graph:

The direction of the graph is due to the fact that the function is a decay function.