Compare Functions

Grade 8 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

📊 Compare Functions

What is it? Comparing functions means analyzing two or more functions to determine their relationships - which grows faster, where they intersect, or which has a greater value at specific points.

Why it's useful: This helps us make real-world decisions like comparing phone plans, predicting which investment grows faster, or determining the better deal between two options.

🔍 How to Compare Functions

  1. Identify the functions - Are they equations, tables, or graphs?
  2. Compare rates of change - Look at slopes or how much y changes when x increases
  3. Compare starting points - Find y-intercepts or initial values
  4. Look for intersections - Where do the functions have the same value?
  5. Make your conclusion - Which is steeper? Which starts higher? When does one overtake the other?

📝 Worked Examples

Example 1: Comparing Equations

Function A: y = 2x + 1    Function B: y = 3x - 2

Step 1: Compare slopes - Function A slope = 2, Function B slope = 3

Step 2: Compare y-intercepts - Function A starts at (0,1), Function B starts at (0,-2)

Step 3: Find intersection: 2x + 1 = 3x - 2 → 1 + 2 = 3x - 2x → 3 = x

Conclusion: Function B grows faster but starts lower. They intersect at (3,7).

Example 2: Table vs Equation

Table Function: x | 0 | 1 | 2 | 3
y | 4 | 6 | 8 | 10

Equation Function: y = 3x + 2

Step 1: Table shows slope = 2 (y increases by 2 each time), Equation has slope = 3

Step 2: Table starts at (0,4), Equation starts at (0,2)

Conclusion: Equation function grows faster but starts lower.

🚨 Common Mistakes

  • Mixing up slope and y-intercept - Remember: slope = rate of change, y-intercept = starting value
  • Only looking at one point - Compare multiple x-values to see the full picture
  • Forgetting units - In real problems, include units like "$ per month" or "miles per hour"
  • Graph reading errors - Double-check your scale on graphs!

💡 Tips & Tricks

  • SLOPE RULE: Bigger slope = faster growth
  • Starting Point: Higher y-intercept = better initially
  • Catch-up Point: Set functions equal to find where they intersect
  • Visualize: Sketch quick graphs to see the comparison

🎯 Practice Suggestions

  • Create your own function pairs and compare them
  • Find real-world examples: compare streaming service prices, cell phone plans
  • Use graphing calculators or online tools to visualize comparisons
  • Practice with mixed representations - compare equations to tables and graphs
  • Work with a partner and explain your reasoning out loud

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 2³ × 3² = ?

Hint: Remember to calculate exponents before multiplication. For example, evaluate 4² first if you see it in an expression.

Show the answer

Answer: 72

  1. Understand the notation** 2³ means 2 raised to the power of 3, which is 2 × 2 × 2. 3² means 3 raised to the power of 2, which is 3 × 3. **
  2. Calculate 2³** 2 × 2 = 4 4 × 2 = 8 So, 2³ = 8. **
  3. Calculate 3²** 3 × 3 = 9 So, 3² = 9. **
  4. Multiply the results** 8 × 9 = 72. **
  5. Final answer** Therefore, 2³ × 3² = 72.

Let's solve the problem step by step. We are given: 2³ × 3² **

2 √(3² + 4²) = ?

Hint: This involves finding the square root of the sum of two squared numbers. Consider what mathematical relationship this represents.

Show the answer

Answer: 5

  1. Calculate 3² = 9
  2. Calculate 4² = 16
  3. Add the results: 9 + 16 = 25
  4. Find the square root: √25 = 5

The answer is 5.

3 2³ × 3² ÷ 6 = ?

Hint: Evaluate the exponents first, then perform the multiplication and division in order from left to right.

Show the answer

Answer: 12

  1. Write the problem clearly. We have: 2³ × 3² ÷ 6
  2. Calculate the exponents first. 2³ means 2 × 2 × 2 = 8 3² means 3 × 3 = 9 So now we have: 8 × 9 ÷ 6
  3. Perform multiplication and division from left to right. First: 8 × 9 = 72 Then: 72 ÷ 6 = 12
  4. Final answer. The result is 12. Explanation: We followed the order of operations: exponents first, then multiplication and division from left to right. We could also rewrite 6 as 2 × 3 and simplify earlier: 2³ × 3² ÷ (2 × 3) = 2³ × 3² ÷ 2 ÷ 3 = 2^(3-1) × 3^(2-1) = 2² × 3¹ = 4 × 3 = 12 Both methods give the same answer.

Let's solve step by step.

4 2³ × (4² - 3²) = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication.

Show the answer

Answer: 56

  1. Evaluate the exponents inside the parentheses** 4² = 4 × 4 = 16 3² = 3 × 3 = 9 So inside parentheses: 4² - 3² = 16 - 9 = 7 Now the expression becomes: 2³ × 7 --- **
  2. Evaluate 2³** 2³ = 2 × 2 × 2 = 8 So now we have: 8 × 7 --- **
  3. Multiply** 8 × 7 = 56 --- **Final Answer:** 56

Let's solve the problem step by step. We have: 2³ × (4² - 3²) = ? --- **

5 2³ × (5² - 3²) = ?

Hint: Remember to follow the order of operations - handle exponents first, then subtraction inside parentheses, and finally multiplication.

Show the answer

Answer: 128

  1. Calculate the exponents: 2³ = 8 and 5² = 25
  2. Calculate the subtraction inside parentheses: 25 - 9 = 16
  3. Multiply the results: 8 × 16 = 128

The answer is 128.

6 √(64) + 2³ × 3 = ?

Hint: Remember to evaluate exponents and square roots before performing multiplication and addition operations

Show the answer

Answer: 32

  1. Evaluate the square root: √(64) = 8
  2. Evaluate the exponent: 2³ = 2 × 2 × 2 = 8
  3. Multiply: 8 × 3 = 24
  4. Add: 8 + 24 = 32

The answer is 32.

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