Exercise 10 Page 407

We want to compare the graphs of an exponential decay function $y=b^x$ and a scatter plot consisting of the given points. To do so, we will draw both graphs.

Graphing an Exponential Decay Function

Since we have an exponential decay function, the value of $b$ must be greater than $0$ and less than $1.$ For simplicity, let's consider $b=0.5.$ The graph of $y=0.5^x$ decreases as $x$ increases, and passes through the points $(0,1)$ and $(1,0.5).$ Also, its horizontal asymptote is the line $y=0.$

Graphing a Scatter Plot

We will make a table of values for the first five points of the scatter plot. Let's use the same value, $b=0.5.$

Point	Substitute	Simplify
$\left(1,b^1\right)$	$\left(1,{0.5}^1\right)$	$(1,0.5)$
$\left(2,b^1+b^2\right)$	$\left(2,{0.5}^1+{0.5}^2\right)$	$(2,0.75)$
$\left(3,b^1+b^2+b^3\right)$	$\left(3,{0.5}^1+{0.5}^2+{0.5}^3\right)$	$(3,0.875)$
$\left(4,b^1+b^2+b^3+b^4\right)$	$\left(4,{0.5}^1+{0.5}^2+{0.5}^3+{0.5}^4\right)$	$(4,0.9375)$
$\left(5,b^1+b^2+b^3+b^4+b^5\right)$	$\left(5,{0.5}^1+{0.5}^2+{0.5}^3+{0.5}^4+{0.5}^5\right)$	$(5,0.96875)$

Now, let's plot these points.

We see that the graph contains the point $(1,0.5).$ The graph is increasing. Moreover, it appears that the line $y=1$ is a horizontal asymptote.

Comparison

Let's examine how both graphs change as $b$ changes. Considering the above graph, we can make a few conclusions about the similarities between the graphs.

• Both contain the point $(1,b).$
• Both have a horizontal asymptote.

We can also consider the differences between the graphs.

• The graph of the exponential decay function is decreasing.
• The graph of the scatter plot is increasing.
• The scatter plot consists of discrete points.
• The graph of the exponential decay function is continuous.