Introduction
Comparing Two Functions is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with comparing two functions.
What Is Comparing Two Functions?
Comparing Two Functions means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Comparing Two Functions
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: Function \(f\) is given by \(f(x)=4x+1\). Function \(g\) is given by the table: \(g(0)=2,\ g(1)=5,\ g(2)=8\). Which function has the greater rate of change?
| \(x\) | 0 | 1 | 2 |
|---|---|---|---|
| \(g(x)\) | 2 | 5 | 8 |
- A. \(f(x)\)
- B. \(g(x)\)
- C. Both are equal
- D. Cannot be determined
Why it works: The slope of \(f\) is \(4\). For \(g\): \(\Delta y/\Delta x=3/1=3\). Since \(4>3\), \(f\) has the greater rate of change.
Answer: \(f(x)\)
Visual Model 2
Question: Function \(h(x)=-2x+7\) and function \(k\) shown in the table: \((0,7)\), \((1,5)\), \((2,3)\). Which statement is true?
| \(x\) | 0 | 1 | 2 |
|---|---|---|---|
| \(k(x)\) | 7 | 5 | 3 |
- A. \(h\) has a steeper slope than \(k\).
- B. \(k\) has a steeper slope than \(h\).
- C. Both have equal slopes.
- D. The slopes cannot be compared.
Why it works: Slope of \(h\): \(-2\). From \(k\)'s table: \((5-7)/(1-0)=-2\). Both have slope \(-2\).
Answer: Both have equal slopes
Worked Examples
Example 1
Question: Function \(p(x)=\frac{3}{2}x-4\) and the table for function \(q\): \((0,2)\), \((2,5)\), \((4,8)\). Which has a steeper slope?
| \(x\) | 0 | 2 | 4 |
|---|---|---|---|
| \(q(x)\) | 2 | 5 | 8 |
- A. \(p(x)\)
- B. \(q(x)\)
- C. Both have equal slopes.
- D. Cannot be determined.
- Slope of \(p\): \(\frac{3}{2}=1.5\).
- From table \(q\): \((5-2)/(2-0)=3/2=1.5\).
- Both have slope \(1.5\).
Answer: Both have equal slopes
Example 2
Question: Plan A costs \($15\) per month: \(A(m)=15+0.10m\). Plan B shown in table: \((0,$20)\), \((100,$25)\), \((200,$30)\), \((300,$35)\). Which plan has the higher base cost?
| \(m\) | 0 | 100 | 200 | 300 |
|---|---|---|---|---|
| Plan B | $20 | $25 | $30 | $35 |
- A. Plan A
- B. Plan B
- C. Both cost the same.
- D. Cannot be determined.
- Plan A base cost: \($15\).
- Plan B at \(m=0\): \($20\).
- Since \(20>15\), Plan B is higher.
Answer: Plan B
Example 3
Question: Table: \((1,2)\), \((2,6)\), \((3,10)\); and \(z(x)=3x+1\). Which crosses the \(y\)-axis higher?
| \(x\) | 1 | 2 | 3 |
|---|---|---|---|
| \(y\) | 2 | 6 | 10 |
- A. Table function
- B. \(z(x)\)
- C. Both have same \(y\)-intercept.
- D. Cannot be determined.
- Table slope: \((6-2)/(2-1)=4\).
- From \((1,2)\): \(2=4(1)+b \Rightarrow b=-2\).
- So \(y\)-intercept is \(-2\).
- For \(z\): \(y\)-intercept is \(1\).
Answer: \(z(x)\)
Real-World Word Problems
Problem 1
Question: Function \(A(t)=60t\) (mph). Function \(B\) passes through \((0,1)\) and \((5,5)\) (on distance-time graph, with \(t\) in hours). Which travels faster?
- A. Function \(A\)
- B. Function \(B\)
- C. Both same speed.
- D. Cannot be determined.
Why it works: Speed is the slope. \(A\) has slope \(60\) mph. \(B\) has slope \((5-1)/(5-0)=4/5=0.8\) mph. Since \(60>0.8\), \(A\) travels faster.
Answer: Function \(A\)
Problem 2
Question: \(S(m)=500+25m\) and table \((0,$600)\), \((3,$675)\), \((6,$750)\). Which starts with more money?
| \(m\) | 0 | 3 | 6 |
|---|---|---|---|
| Table function | $600 | $675 | $750 |
- A. \(S\)
- B. Table function
- C. Both same.
- D. Cannot be determined.
Why it works: \(S(0)=500\). Table shows \(600\) at \(m=0\). Since \(600>500\), table function starts higher.
Answer: Table function
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Function \(r\) passes through \((1,3)\) and \((3,7)\). Function \(s(x)=x+3\). Which increases faster?
- A. Function \(r\)
- B. Function \(s\)
- C. Both increase equally.
- D. Cannot be determined.
Question 2
Function \(u\) passes through \((0,-1)\) and \((2,3)\). Function \(v(x)=-x+5\). Which has a positive slope?
- A. Only \(u\)
- B. Only \(v\)
- C. Both \(u\) and \(v\)
- D. Neither \(u\) nor \(v\)
Question 3
Function \(f\): "increases at 1.5 per unit, starts at 20." Function \(g\): "increases at 2 per unit, starts at 25." Which function is larger at \(t=10\)?
- A. Function \(f\)
- B. Function \(g\)
- C. Both equal at \(t=10\)
- D. Cannot be determined.
Question 4
\(f\) passes through \((2,5)\) and \((4,1)\). \(g(x)=-x+8\). Which has a more negative slope?
- A. \(f(x)\)
- B. \(g(x)\)
- C. Both equal.
- D. Not comparable.
Question 5
Company A: \(A(d)=40+0.15d\). Company B through \((100,55)\) and \((200,80)\). Which has higher rate?
- A. Company A
- B. Company B
- C. Both same.
- D. Cannot be determined.
Question 6
Plant \(X\): \(H(d)=2d+10\) cm. Plant \(Y\) at days \(0, 5, 10\) has heights \(15, 20, 25\) cm. Which started taller?
- A. Plant \(X\)
- B. Plant \(Y\)
- C. Both same.
- D. Cannot be determined.
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: Function \(r\)
Slope of \(r\): \((7-3)/(3-1)=4/2=2\). Slope of \(s\): \(1\). Since \(2>1\), \(r\) increases faster.
Question 2
Answer: Only \(u\)
Slope of \(u\): \((3-(-1))/(2-0)=4/2=2>0\). Slope of \(v\): \(-1<0\). Only \(u\) is positive.
Question 3
Answer: Function \(g\)
\(f(10)=20+1.5(10)=35\). \(g(10)=25+2(10)=45\). Since \(45>35\), \(g\) is larger at \(t=10\).
Question 4
Answer: \(f(x)\)
Slope of \(f\): \((1-5)/(4-2)=-4/2=-2\). Slope of \(g\): \(-1\). Comparing: \(-2\) is further left on the number line than \(-1\), so \(f\) with slope \(-2\) is more negative than \(g\) with slope \(-1\).
Question 5
Answer: Company B
\(A\) rate: \(0.15\). \(B\) rate: \((80-55)/(200-100)=25/100=0.25\). Since \(0.25>0.15\), \(B\) is higher.
Question 6
Answer: Plant \(Y\)
\(X\) at day 0: \(10\) cm. \(Y\) at day 0: \(15\) cm. Since \(15>10\), \(Y\) started taller.
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Comparing Two Functions becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
