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

4.5.3 Comparing Functions

4.5.3 • Comparing Functions

Activity

A college student is choosing between two data plans for her new cell phone. Both plans include an allowance of 2 gigabytes of data per month. The monthly cost of each option can be seen as a function and represented with an equation:

Option A: $A (x) = 60$
Option B: $B (x) = 10 x + 25$

In each function, the input, $x$ , represents the gigabytes of data used over the monthly allowance.

The student decides to find the cost of each option when she exceeds the 2 gigabytes of data allowed each month. Use the function rules to help her compare the data plans.

1. What is the value of A(1)

60.

2. What is the value of B(1)

35

3. Explain which data plan seems to be the best option for when 1 additional gigabyte of data is used.

Compare your answer:

If only 1 extra gigabyte is used, then Option B is the best option because it would be cheaper.

4. What is the value of $A (7.5)$ ?

60.

$A (7.5) = 60$

5. What is the value of $B (7.5)$ ?

100.

$B (7.5) = 10 (7.5) + 25 = 100$

6. Explain which data plan seems to be the best option for when 7.5 additional gigabytes of data are used.

Compare your answer:

If 7.5 extra gigabytes are used, then Option A is the best option because it would be cheaper.

7. Describe option A, $A (x) = 60$ , in words.

Compare your answer:

Your answer may vary, but here is a sample: Option A charges a flat fee of $60 each month for any data used over 2 gigabytes. It doesn't matter how many gigabytes of data are used.

8. Describe option B, $B (x) = 10 x + 25$ , in words.

Compare your answer:

Your answer may vary, but here is a sample: Option B charges a $25 fee plus $10 for each gigabyte of data over the 2-gigabytes allowance.

9. Use the desmos graphing tool or technology outside the course. Graph each function on the same coordinate plane.

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10. Explain which plan you think she should choose.

Compare your answer:

Your answer may vary, but here are some samples: She should choose Option A if she plans on using around 7.5 gigabytes a month. She should choose Option A if she uses more than 5.5 gigabytes a month (or more than 3.5 gigabytes over the allowance). She should choose Option B if she uses only about 2 gigabytes a month.

11. The student only budgeted $50 a month for her cell phone. She thought, “I wonder how many gigabytes of data I would have for $50 if I go with Option B?” and wrote $B (x) = 50$ . How many additional gigabytes can the student use beyond the 2 allotted by the plan?

Compare your answer:

She can use 2.5 gigabytes beyond the initial 2 provided by the plan (for a total of 4.5 gigabytes each month).

12. Explain how you knew how many gigabytes of data she would have for $50.

Compare your answer:

Your answer may vary, but here are some samples: I used the graph to estimate the gigabytes of data when the cost is $50, and it is about 2.5. I knew that $B (x)$ is equal to $10 x + 25$ , so I wrote $10 x + 25 = 50$ and solved for $x$ , which gave $x = 2.5$ .

Are you ready for more?

Extending Your Thinking

Describe a different data plan that, for any amount of data used, would cost no more than one of the given plans and no less than the other given plan. Explain or show how you know this data plan would meet these requirements.

Your answers may vary, but here are some examples:

The graph of this data plan would be a line whose slope is between the slope of the other two graphs (greater than 0 but less than 10) and goes through the intersection of the other two graphs.

The plan charges $4 for each gigabyte of data over the 2-gigabyte allowance, plus a $6 fee. With this plan:

If we use exactly 13.5 additional gigabytes above the allowance (that is, 5.5 gigabytes in total), all three plans would cost $60.
If we use less than that, our plan would cost less than Option A but more than Option B.
If we use more than that, our plan would cost more than Option A but less than Option B.

Self Check

Rental car company A charges $50 a day plus $0.50 a mile,

m

, and is modeled by function

A

.

Company A is always more than Company B.
$A(100)<B(100)$
$A(100)=B(100)$
$A(100)>B(100)$

Additional Resources

Comparing Function Values

Functions are used in a variety of real-life situations. They are useful in making comparisons, especially when comparing the cost.

Example

The daily cost to printing company A to print a book is modeled by a function $C (x) = 3.25 x + 1500$ , where $C$ is the total daily cost, in dollars, and $x$ is the number of books printed.

Their competitor, company B, can print a book at a cost modeled by $C (x) = 2.75 x + 2500$ .

Find $C (1000)$ for each company and make a statement about the result.

Company A: $C (1000) = 3.25 (1000) + 1500$

$C = 4750$

Company B: $C (1000) = 2.75 (1000) + 2500$

$C = 5250$

When 1,000 books are printed, Company A has the better price.

Try it

Comparing Function Values

The daily cost to manufacturing company A is modeled by the function $C (x) = 7.25 x + 2500$ , where $C (x)$ is the total daily cost and $x$ is the number of items manufactured.

The daily cost to manufacturing company B is modeled by the function $C (x) = 8.5 x + 1500$ .

Which company is less at $C (1000)$ ? Explain the meaning of the result.

Your answers may vary, but here are some examples:

Here is how to determine which company costs less at $C (1000)$ :

Substitute 1,000 in for $x$ in each function.

Company A’s cost equation

C (x) = 7.25 x + 2500

Substitute $x = 1000$

C (1000) = 7.25 (1000) + 2500

Simplify

C (1000) = 9, 750

Company B’s cost equation

C (x) = 8.5 x + 1500

Substitute $x = 1000$

C (1000) = 8.5 (1000) + 1500

Simplify

C (1000) = 10, 000

When 1,000 items are manufactured, Company A costs less.