A.10(F)·Algebra II·Rational FunctionsReadiness

Modeling Average Cost

Write and interpret rational functions in real-world situations.

Students build average-cost models by dividing a total-cost rule by the number of items, then interpret how a one-time fee is spread over more items.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Total cost first

A one-time fee plus a constant cost per item gives a linear total-cost function: $C(x)=\text{rate}\cdot x+\text{fee}$ , where x is the number of items. Include taxes or other charges only when the context says to add them.

Divide for the average

Average cost per item is $a(x)=\frac{C(x)}{x}$ . Because x is in the denominator, the model is rational. In context, x must be a positive whole number.

Long-run behavior

Rewrite $\frac{\text{rate}\cdot x+\text{fee}}{x}$ as $\text{rate}+\frac{\text{fee}}{x}$ . As x becomes large, the fee's share shrinks, so the average approaches the per-item rate. That rate is a horizontal asymptote.

Worked example

Custom T-shirts cost 6 dollars each plus a 120-dollar setup fee. How many shirts make the average cost 10 dollars per shirt?

Total cost: $C(x)=6x+120$ .
Average cost: $a(x)=\frac{6x+120}{x}$ .
Set $\frac{6x+120}{x}=10$ and multiply by x: $6x+120=10x$ .
Solve: $120=4x$ , so $x=30$ . Check: $\frac{300}{30}=10$ .

30 shirts

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Solving Contextual Rational Equations

UT IPSI · Kid2Kid / Texas Education Agency

Model real-world situations with rational equations and interpret their solutions.

Watch on YouTube

Texas public school video

Algebra 2 Modeling With Direct or Inverse Variation

UT IPSI · Kid2Kid / Texas Education Agency

Recognize quantities that vary inversely, a key relationship in rational models.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. A gym charges a 50-dollar joining fee plus 25 dollars per month. Write the average monthly cost for m months.
Show answer and reasoning
$a(m)=\frac{25m+50}{m}$
The total cost is 25m + 50 dollars. Dividing by the number of months gives the average cost per month.
2. For $a(x)=\frac{4x+200}{x}$ , what value does the average cost approach when x is very large?
Show answer and reasoning
4 dollars per item
Rewrite as $4+\frac{200}{x}$ . The fraction approaches 0 as x grows, leaving the per-item cost.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 5 · 1 point

The cost of printing cookbooks is $2.09 per book plus a one-time setup fee of $349. All taxes are included. Which function models $a(x)$ , the average cost per book of printing $x$ cookbooks?

$a(x) = 2.09x + \frac{349}{x}$
$a(x) = \frac{2.09x + 349}{x}$
$a(x) = \frac{2.09 + 349}{2.09x}$
$a(x) = \frac{2.09x + 349}{2.09x}$

Show STAAR answer and explanation

$a(x) = \frac{2.09x + 349}{x}$

The total cost is $2.09x+349$ . Average cost per book divides that total by the number of books, giving $a(x)=\frac{2.09x+349}{x}$ .

$a(x) = 2.09x + \frac{349}{x}$ : The per-book cost should be 2.09, not $2.09x$ ; the full total-cost numerator must be divided by x.

$a(x) = \frac{2.09 + 349}{2.09x}$ : The numerator must include the variable printing cost $2.09x$ , and average cost divides by x alone.

$a(x) = \frac{2.09x + 349}{2.09x}$ : Dividing by $2.09x$ introduces an extra 2.09 factor; the number of books is only x.

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Archived released STAAR Algebra II EOC · 2013 · Item 28 · 1 point

The function $R = \frac{9x}{9 + x}$ models the equivalent resistance in ohms, $R$ , when a fixed 9-ohm resistor is connected in parallel with a variable resistor of $x$ ohms. Which value of $x$ is closest to the number of ohms needed in the variable resistor for there to be an equivalent resistance, $R$ , of 6.4 ohms?

15.4
57.6
3.7
22.2

Show STAAR answer and explanation

22.2

Set $6.4=\frac{9x}{9+x}$ and multiply: $57.6+6.4x=9x$ . Then $2.6x=57.6$ , so $x\approx22.2$ ohms.

15.4: 15.4 is $9+6.4$ . Substituting it gives $R\approx5.7$ ohms.

57.6: 57.6 is $6.4(9)$ , an intermediate value. Divide by 2.6 to finish solving.

3.7: $x=3.7$ gives $R\approx2.6$ ohms. A parallel resistance is less than either resistor, so x must be greater than 6.4.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Write a total-cost function with a fixed fee and a per-item cost.
Write the average-cost rational function by dividing total cost by the number of items.
Interpret the long-run behavior of average cost and solve for a target average.

Instructional sequence

1
Engage
5 min
A club pays a 100-dollar setup fee plus 2 dollars per shirt. Ask whether ordering more shirts makes each one cheaper on average.
2
Direct instruction
10 min
Separate total cost from average cost. Model C(x) = fixed + rate·x and a(x) = C(x)/x.
3
Table investigation
13 min
Compute average cost for 10, 50, 100, and 1,000 items and describe the trend toward the per-item rate.
4
Solve in context
15 min
Set the average-cost function equal to a target, multiply both sides by x, and interpret the solution as a number of items.
5
Closure
7 min
Students explain the meaning of a(x) and complete the exit ticket.

Misconceptions to address

““The average cost function is the same as total cost.””

Teacher move: Total cost answers “how much altogether?” Average cost answers “how much per item?” Divide the whole total by x.

““Only the setup fee is divided by x.””

Teacher move: Average cost is $\frac{\text{rate}\cdot x+\text{fee}}{x}$ . Rewriting gives $\text{rate}+\frac{\text{fee}}{x}$ , so the per-item cost remains.

Materials

•Printing and event pricing scenarios
•Average-cost table template
•Graphing calculator

Teaching tips

Write “total ÷ number of items” above every average-cost model.
Use a table before the graph so students see the fixed fee spreading out.
Require a units sentence such as “dollars per book” for every interpreted value.

Exit ticket · 3–5 min

A recording studio charges a 300-dollar setup fee plus 15 dollars per CD. Write the average-cost function and find the average cost for 20 CDs.

Show expected answer

$a(x)=\frac{15x+300}{x}$ . For 20 CDs, $a(20)=\frac{600}{20}=30$ dollars per CD.

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Total cost first

Divide for the average

Long-run behavior

Worked example

Watch the lesson

Algebra 2 Solving Contextual Rational Equations

Algebra 2 Modeling With Direct or Inverse Variation

Try it yourself

1. A gym charges a 50-dollar joining fee plus 25 dollars per month. Write the average monthly cost for m months.

2. For a(x)=4x+200xa(x)=\frac{4x+200}{x}a(x)=x4x+200​, what value does the average cost approach when x is very large?

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 5 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 28 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Table investigation

Solve in context

Closure

Misconceptions to address

2. For $a(x)=\frac{4x+200}{x}$ , what value does the average cost approach when x is very large?