Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

5.3.3 Compare and Contrast Two Patterns

5.3.3 • Compare and Contrast Two Patterns

Activity

Here are graphs showing how the amount of money in each of the purses change. Remember, Purse A starts with $1,000 and grows by $200 each day. Purse B starts with $0.01 and doubles each day.

1. Which data on the graph shows the amount of money in Purse $A$ ? Which data on the graph shows the amount of money in Purse $B$ ? Explain how you know.

Compare your answer:

The blue data on the graph containing $P$ is Purse $B$ , and the red data on the graph containing $Q$ is Purse $A$ . Purse $B$ starts with less money than Purse $A$ , so its graph will be lower at the beginning.The blue data on the graph containing $P$ is Purse $B$ , and the red data on the graph containing $Q$ is Purse $A$ . Purse $B$ starts with less money than Purse $A$ , so its graph will be lower at the beginning.

2. Points $P$ and $Q$ are labeled on the graph. Explain what they mean in terms of the genie’s offer.

Compare your answer:

Point $P$ says that 9 days after the genie has appeared, Purse B contains $5.12. Point $Q$ says that 5 days after the genie has appeared, Purse A contains $2,000.

3. What are the coordinates of the vertical intercept for each graph? Explain how you know.

Compare your answer:

For Purse A, the vertical intercept is $(0, 1, 000)$ because when the genie appears (day 0), the purse has $1,000. For Purse B, the vertical intercept is $(0, 0.01)$ because it only contains $0.01 when the genie appears.

4. When does Purse B become a better choice than Purse A? Be prepared to show your reasoning.

Compare your answer:

According to the graphs, on day 19 and afterward, Purse B is worth more than Purse A. This can be seen on the graph since Purse B has a greater $y$ -coordinate for the first time when $x$ is 19.

5. Knowing what you know now, which purse would you choose? Be prepared to show your reasoning.

Compare your answer:

Your answer may vary, but here is a sample: Purse $B$ is worth more after 19 days, but there may be other considerations to take into account.

While on a beach, your friend discovers a different genie. This genie also offers two purses to choose from, and he gives you the following graph to show how the money in each purse will grow. The set of triangular marks that lie on a line represent values in Purse A, and the set of square marks that make a curve represent those in Purse B.

6. The genie is still deciding how many days he will let the money in the purses grow. Help your friend plan a strategy for picking the purse with more money while the genie thinks.

Your answers may vary, but here is an example:

The vertical intercept for Purse B is lower, so Purse B starts off with less money than Purse A. The values in Purse B stay lower than the values in Purse A until day 10, when they appear to be the same. So Purse A is better up to day 10, and after that point Purse B becomes a better choice.

Are you ready for more?

Extending Your Thinking

“Okay, okay,” the genie smiles, disappointed. “I will give you an even more enticing deal.” He explains that Purse B stays the same, but Purse A now increases by $250,000 every day. Which purse should you choose?

Your answers may vary, but here is an example:

Purse B surpasses Purse A on day 30, with about $10.7 million in Purse B and only about $7.5 million in Purse A. So it’s better to choose Purse A if the amounts grow for less than 30 days, and it’s better to choose Purse B if they grow for 30 days or more.

Self Check

The graph shows the yearly balance, in dollars, in two investment accounts. Account A starts with $2,000. Account B starts with $1,000.

Both accounts grow by the same amount each year.
Both accounts grow more quickly as time goes on.
Account A grows by the same amount each year. Account B grows more quickly as time goes on
Account A grows more quickly as time goes on. Account B grows by the same amount each year.

Additional Resources

Compare and Contrast Graphs

Compare and contrast the graphs of $y = 2 x$ and $y = 2^{x}$ .

To compare and contrast the graphs of these functions, you would first need to graph them.

$x$	$y = 2 x$
–1	–2
0	0
1	2
2	4

Graph of

(y = 2 x)

$x$	$y = 2^{x}$
–1	$\frac{1}{2}$
0	1
1	2
2	4

Graph of

(y = 2^{x})

Both equations have a 2 and an $x$ in them, but in $y = 2 x$ , $x$ is multiplied by 2, and in $y = 2^{x}$ , 2 is the base and $x$ is the exponent. If you graph the equations together on the same grid, it makes it easier to discuss the key features and compare the graphs.

The graphs intersect at $(1, 2)$ and $(2, 4)$ .
$y = 2 x$ has a $y$ -intercept of $(0, 0)$ . $y = 2^{x}$ has a $y$ -intercept of $(0, 1)$ .
When $1 < x < 2$ , $y = 2 x$ is greater than $y = 2^{x}$ .

Try it

Compare and Contrast Graphs

Compare and contrast the graphs of $y = 3 x + 1$ and $y = 3^{x}$ .

Compare your answer:

Here is how to compare and contrast the graphs of $y = 3 x + 1$ and $y = 3^{x}$ :

First, you need to graph both equations on the same plane.

Then you can compare key features of the graph, like points where they are equal and intercepts. Answers may vary, but here is a sample answer. Your answer may include all or part of the sample.

The functions are equal at their $y$ -intercept $(0, 1)$ and also at $(1.6, 5.8)$ .
$y = 3 x + 1$ is linear and is greater when $0 < x < 1.6$ .
$y = 3^{x}$ is exponential and greater when $x < 0$ and $x > 1.6$ .