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Compare Functions

Grade 8 · Algebra · Worksheet 2

  1. Function A: y = 13x + 4. Function B: table below. Which function has the greater y-intercept? x | y 0 | 18 2 | 44 4 | 70 6 | 96 Answer: ______________
  2. Aroha is comparing two different dog-walking services. Service A charges a flat fee of $5 per walk plus $3 per kilometer walked. Service B is described by the equation y = 5x + 9, where y is the total cost in dollars and x is the number of kilometers walked. Aroha wants to know which service is cheaper for a 7-kilometer walk, and by how much. Which service is cheaper, and what is the cost difference? Answer: ______________
  3. Function A: y = 7x + 5. Function B: table below. Which function has the greater slope? x | y 1 | 13 3 | 29 5 | 45 7 | 61 Answer: ______________
  4. (3² × 4) - (2³ + 5) = ? Answer: ______________
  5. Liam is comparing two phone plans. Plan A costs $25 per month plus $0.10 per text message. Plan B has a fixed cost of $40 per month for unlimited texting. Liam wants to determine how many text messages he would need to send for both plans to cost the same amount. Write an equation to represent this situation and solve for the number of text messages. Answer: ______________
  6. (2.5 × 10³) × (4 × 10⁻²) = ? Answer: ______________
  7. Liam is comparing two different cell phone plans. Plan A charges a flat fee of $20 per month plus $0.10 per minute of talk time. Plan B charges a flat fee of $15 per month plus $0.15 per minute of talk time. Liam wants to know how many minutes of talk time would make both plans cost the same amount. Write an equation to represent this situation and solve for the number of minutes. Answer: ______________
  8. (5 × 10³) × (2 × 10⁴) = ? Answer: ______________
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Answer Key & Explanations

Compare Functions · Grade 8 · Worksheet 2

  1. Function A: y = 13x + 4. Function B: table below. Which function has the greater y-intercept? x | y 0 | 18 2 | 44 4 | 70 6 | 96 Answer: Function B Solution: Find the y-intercept of Function A. In the equation y = 13x + 4, the y-intercept is the constant term, which is 4. Find the y-intercept of Function B.
    Full step-by-step solution

    Step 1: Find the y-intercept of Function A. In the equation y = 13x + 4, the y-intercept is the constant term, which is 4. Step 2: Find the y-intercept of Function B. From the table, when x = 0, y = 18. So the y-intercept of Function B is 18. Step 3: Compare the y-intercepts. Function A has y-intercept 4, Function B has y-intercept 18. Since 18 > 4, Function B has the greater y-intercept. The answer is Function B.

  2. Aroha is comparing two different dog-walking services. Service A charges a flat fee of $5 per walk plus $3 per kilometer walked. Service B is described by the equation y = 5x + 9, where y is the total cost in dollars and x is the number of kilometers walked. Aroha wants to know which service is cheaper for a 7-kilometer walk, and by how much. Which service is cheaper, and what is the cost difference? Answer: Service A is cheaper by $3 Solution: Write the equation for Service A. Flat fee = $5, rate per km = $3. So equation: y = 3x + 5.
    Full step-by-step solution

    Step 1: Write the equation for Service A. Flat fee = $5, rate per km = $3. So equation: y = 3x + 5. Step 2: Service B equation: y = 5x + 9. Step 3: For a 7 km walk (x = 7): Service A: y = 3(7) + 5 = 21 + 5 = 26 dollars. Service B: y = 5(7) + 9 = 35 + 9 = 44 dollars. Step 4: Compare: 26 < 44, so Service A is cheaper. Step 5: Difference = 44 - 26 = 18 dollars. The answer is Service A is cheaper by $18.

  3. Function A: y = 7x + 5. Function B: table below. Which function has the greater slope? x | y 1 | 13 3 | 29 5 | 45 7 | 61 Answer: Function A Solution: Find the slope of Function A. In the equation y = 7x + 5, the slope is the coefficient of x, which is 7. Find the slope of Function B.
    Full step-by-step solution

    Step 1: Find the slope of Function A. In the equation y = 7x + 5, the slope is the coefficient of x, which is 7. Step 2: Find the slope of Function B. Use two points from the table, such as (1, 13) and (3, 29). Slope = (29 - 13) / (3 - 1) = 16 / 2 = 8. Step 3: Compare the slopes. Function A has slope 7, Function B has slope 8. Since 8 > 7, Function B has the greater slope. The answer is Function B.

  4. (3² × 4) - (2³ + 5) = ? Answer: 23 Solution: We have: (3² × 4) - (2³ + 5) Handle exponents first (order of operations: PEMDAS/BODMAS). 3² means 3 × 3 = 9 2³ means 2 × 2 × 2 = 8 (9 × 4) - (8 + 5) Simplify inside parentheses.
    Full step-by-step solution

    Let's solve step by step. We have: (3² × 4) - (2³ + 5) Step 1: Handle exponents first (order of operations: PEMDAS/BODMAS). 3² means 3 × 3 = 9 2³ means 2 × 2 × 2 = 8 So the expression becomes: (9 × 4) - (8 + 5) Step 2: Simplify inside parentheses. 9 × 4 = 36 8 + 5 = 13 Now we have: 36 - 13 Step 3: Perform the subtraction. 36 - 13 = 23 Final answer: 23

  5. Liam is comparing two phone plans. Plan A costs $25 per month plus $0.10 per text message. Plan B has a fixed cost of $40 per month for unlimited texting. Liam wants to determine how many text messages he would need to send for both plans to cost the same amount. Write an equation to represent this situation and solve for the number of text messages. Answer: 150 Solution: Plan A cost = 25 + 0.10 * x Plan B cost = 40 25 + 0.10 * x = 40 Subtract 25 from both sides to isolate the term with x. 0.10 * x = 40 - 25 0.10 * x = 15 Divide both sides by 0.10 to solve for x.
    Full step-by-step solution

    Let's define the number of text messages as x. Plan A cost = 25 + 0.10 * x Plan B cost = 40 We want the costs to be equal: 25 + 0.10 * x = 40 Step 1: Subtract 25 from both sides to isolate the term with x. 0.10 * x = 40 - 25 0.10 * x = 15 Step 2: Divide both sides by 0.10 to solve for x. x = 15 / 0.10 Step 3: Calculate the division. 15 divided by 0.10 is the same as 15 divided by (1/10), which is 15 * 10 = 150. So, x = 150. This means Liam would need to send 150 text messages for both plans to cost the same.

  6. (2.5 × 10³) × (4 × 10⁻²) = ? Answer: 100 Solution: Multiply the coefficients: 2.5 × 4 = 10 Add the exponents: 3 + (-2) = 1 Combine the results: 10 × 10¹ = 10 × 10 = 100 The answer is 100.
    Full step-by-step solution

    Step 1: Multiply the coefficients: 2.5 × 4 = 10 Step 2: Add the exponents: 3 + (-2) = 1 Step 3: Combine the results: 10 × 10¹ = 10 × 10 = 100 The answer is 100.

  7. Liam is comparing two different cell phone plans. Plan A charges a flat fee of $20 per month plus $0.10 per minute of talk time. Plan B charges a flat fee of $15 per month plus $0.15 per minute of talk time. Liam wants to know how many minutes of talk time would make both plans cost the same amount. Write an equation to represent this situation and solve for the number of minutes. Answer: 100 Solution: Plan A cost = flat fee + cost per minute × minutes Plan A cost = 20 + 0.10 × m Plan B cost = flat fee + cost per minute × minutes Plan B cost = 15 + 0.15 × m We want the number of minutes where both plans cost the same, so we set the costs equal: 20 + 0.10m = 15 + 0.15m Now solve for m.
    Full step-by-step solution

    Let's define the variable m as the number of minutes of talk time. Plan A cost = flat fee + cost per minute × minutes Plan A cost = 20 + 0.10 × m Plan B cost = flat fee + cost per minute × minutes Plan B cost = 15 + 0.15 × m We want the number of minutes where both plans cost the same, so we set the costs equal: 20 + 0.10m = 15 + 0.15m Now solve for m. Step 1: Subtract 15 from both sides 20 - 15 + 0.10m = 0.15m 5 + 0.10m = 0.15m Step 2: Subtract 0.10m from both sides 5 = 0.15m - 0.10m 5 = 0.05m Step 3: Divide both sides by 0.05 m = 5 / 0.05 m = 500 / 5 m = 100 So at 100 minutes of talk time, both plans cost the same amount.

  8. (5 × 10³) × (2 × 10⁴) = ? Answer: 100000000 Solution: Multiply the coefficients: 5 × 2 = 10 Add the exponents: 3 + 4 = 7 Write in scientific notation: 10 × 10⁷ Convert to standard form: 10 × 10,000,000 = 100,000,000 The answer is 100,000,000
    Full step-by-step solution

    Step 1: Multiply the coefficients: 5 × 2 = 10 Step 2: Add the exponents: 3 + 4 = 7 Step 3: Write in scientific notation: 10 × 10⁷ Step 4: Convert to standard form: 10 × 10,000,000 = 100,000,000 Step 5: The answer is 100,000,000