Graphs Representing Exponential Decay

5.12.3 Graphs Representing Exponential Decay

5.12.3 • Graphs Representing Exponential Decay

Activity

Classroom Connection

Log into student.desmos.com using the information provided by your teacher to complete the activity.

1. Examine the graph of the functions below.

Match each of the functions with a graph. Be prepared to explain your reasoning.

a. Which graph best represents $m (x) = 200 \cdot (\frac{1}{4})^{x}$ ?

Graph D

b. Which graph best represents $n (x) = 200 \cdot (\frac{1}{2})^{x}$ ?

Graph C

c. Which graph best represents $p (x) = 200 \cdot (\frac{3}{4})^{x}$ ?

Graph B

d. Which graph best represents $q (x) = 200 \cdot (\frac{7}{8})^{x}$ ?

Graph A

2. Functions $f$ and $g$ are defined by these equations:

$f (x) = 1, 000 \cdot (\frac{1}{10})^{x}$
$g (x) = 1, 000 \cdot (\frac{9}{10})^{x}$

a. Use the graphing tool or technology outside the course. On the same coordinate plane, sketch a graph that represents the functions, $f$ and $g$ . It may be helpful to use a different color for each function.

Access multimedia content

Compare your answers:

Line graph with two curves. Curve f (orange) decreases steeply and approaches zero quickly. Curve g (teal) decreases more gradually. Both curves have labeled axes: x (horizontal) and y (vertical).

b. Which function is decaying more quickly?

$f (x)$

c. Explain your reasoning regarding which function decays more quickly.

Compare your answers:

$f (x) = 1, 000 \cdot (\frac{1}{10})^{x}$ has a smaller base than $g (x) = 1, 000 \cdot (\frac{9}{10})^{x}$ . Both bases are less than 1, so both functions decrease, but $f (x) = 1, 000 \cdot (\frac{1}{10})^{x}$ decreases more rapidly.

Building Character: Grit

A person in business attire climbs a mountain with a briefcase and walking stick. Trophy and bullseye icons appear in the background, symbolizing achievement and reaching goals. Orange flags and clouds complete the scene.

Grit predicts accomplishing challenging goals of personal significance. For example, grittier students are more likely to graduate from high school, and grittier cadets are more likely to complete their training at West Point.

Think about your current sense of grit. Are the following statements true for you?

There is at least one subject or activity that I never get bored of thinking about.
I finish whatever I begin.

Don’t worry if none of these statements are true for you. Developing this trait takes time. Your first step starts today!

Self Check

Which of the following functions will decay the fastest?

$j(x)=0.5^x$
$h(x)=1.34^x-90$
$g(x)=0.34^x+2$
$f(x)=0.86^x-10$

Additional Resources

Matching Exponential Functions and Graphs

For each of the following, match each function with one of the graphs in the picture below.

Six labeled curves (A–F) cross at the origin on an x-y axis. Curves A and B open left, C and D open up, E and F open right. Each curve is a different color and labeled near its end.

Remember, the equation for an exponential function is $y = a \cdot b^{x}$ . The $a$ will affect the $y$ -intercept. When $b < 1$ , there is exponential decay. The closer $b$ is to 0, the faster it decays. The closer $b$ is to 1, the slower it decays. When $b > 1$ , the larger the value of $b$ , the faster it grows.

Equation	$y$ -intercept	Growth/Decay	Match
1. $f (x) = 2 (0.68)^{x}$	$(0, 2)$	Decay	B
2. $f (x) = 2 (1.28)^{x}$	$(0, 2)$	Growth	F
3. $f (x) = 2 (0.81)^{x}$	$(0, 2)$	Decay	A
4. $f (x) = 4 (1.28)^{x}$	$(0, 4)$	Growth	D
5. $f (x) = 2 (1.59)^{x}$	$(0, 2)$	Growth	E
6. $f (x) = 4 (0.68)^{x}$	$(0, 4)$	Decay	C

Try it

Try It: Matching Exponential Functions and Graphs

In the graph above, which function, $f (x) = a \cdot b^{x}$ , has the largest value for $a$ ?

Compare your answer:

Here is how to identify which has the largest value for $a$ :

In the function $f (x) = a \cdot b^{x}$ , $a$ changes the $y$ -intercept. The function with the largest $y$ -intercept is C. It crosses the $y$ -axis at the highest point.