Compare Functions

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Compare Functions · Grade 8 · Worksheet 1

1. 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.

2. Function A: y = 4x + 6. Function B: table below. Which function has the greater y-intercept? x | y 0 | 10 2 | 14 4 | 18 6 | 22 Answer: Function B Solution: Find the y-intercept of Function A. In the equation y = 4x + 6, the y-intercept is the constant term, which is 6. 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 = 4x + 6, the y-intercept is the constant term, which is 6. Step 2: Find the y-intercept of Function B. In the table, when x = 0, y = 10. So the y-intercept of Function B is 10. Step 3: Compare the y-intercepts. Function A has y-intercept 6, Function B has y-intercept 10. Since 10 > 6, Function B has the greater y-intercept. The answer is Function B.

3. Isabella is comparing two different dog-walking services. Woof Walks is described by the equation y = 7x + 12, where y is the total cost in dollars and x is the number of walks. Paws & Claws has a table of costs: for 2 walks the cost is $22, for 4 walks the cost is $37, for 6 walks the cost is $52, and for 8 walks the cost is $67. Isabella wants to know which service has a lower cost per walk and which has a lower initial fee. Which service has the lower cost per walk, and what is that cost per walk? Answer: Paws & Claws, $7.50 per walk Solution: Analyze Woof Walks from the equation y = 7x + 12. The cost per walk (rate) is $7, and the initial fee (y-intercept) is $12. Analyze Paws & Claws from the table.
   Full step-by-step solution

   Step 1: Analyze Woof Walks from the equation y = 7x + 12. The cost per walk (rate) is $7, and the initial fee (y-intercept) is $12. Step 2: Analyze Paws & Claws from the table. Choose two points: (2, 22) and (4, 37). The rate of change (cost per walk) = (37 - 22) / (4 - 2) = 15 / 2 = $7.50 per walk. Step 3: Check the rate with another pair: (4, 37) and (6, 52). Rate = (52 - 37) / (6 - 4) = 15 / 2 = $7.50 per walk. This confirms the rate is constant. Step 4: Find the initial fee for Paws & Claws by working backwards. Using the point (2, 22) and rate of $7.50: Initial fee = 22 - (2 * 7.50) = 22 - 15 = $7. Step 5: Compare rates: Woof Walks has $7 per walk, Paws & Claws has $7.50 per walk. The lower cost per walk is $7 (Woof Walks). But the question asks for the service with the lower cost per walk, which is Woof Walks at $7 per walk. However, reviewing the question carefully: 'Which service has the lower cost per walk, and what is that cost per walk?' The answer is Woof Walks, $7 per walk. The answer is Woof Walks, $7 per walk.

4. Function A: y = 7x + 3. Function B: table below. Which function has the greater slope? x | y 1 | 13 3 | 31 5 | 49 7 | 67 Answer: Function B Solution: Find the slope of Function A. In the equation y = 7x + 3, 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 + 3, the slope is the coefficient of x, which is 7. Step 2: Find the slope of Function B. Use two points from the table, for example (1, 13) and (3, 31). Slope = (31 - 13) / (3 - 1) = 18 / 2 = 9. Step 3: Compare the slopes. Function A has slope 7, Function B has slope 9. Since 9 > 7, Function B has the greater slope. The answer is Function B.

5. 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.

6. Hana is comparing two different car rental companies for her family holiday. Company A charges a fixed booking fee of $60 plus $0.35 per kilometre driven. Company B's charges are shown in the table below for the number of kilometres driven and the total cost. Hana plans to drive 400 kilometres. Which company offers the lower total cost for 400 kilometres, and by how much? Answer: Company A is cheaper by $5 Solution: Write the equation for Company A. Cost = 0.35 per km plus $60 fixed fee. So y = 0.35x + 60.
   Full step-by-step solution

   Step 1: Write the equation for Company A. Cost = 0.35 per km plus $60 fixed fee. So y = 0.35x + 60. Step 2: Find the rate for Company B from the table. Choose two points: (100, 95) and (200, 140). Rate = (140 - 95) / (200 - 100) = 45 / 100 = 0.45 dollars per km. Step 3: Find the fixed fee for Company B. Use point (100, 95) and rate 0.45: 95 = 0.45(100) + b → 95 = 45 + b → b = 50. So Company B's equation is y = 0.45x + 50. Step 4: Calculate cost for 400 km with Company A: y = 0.35(400) + 60 = 140 + 60 = $200. Step 5: Calculate cost for 400 km with Company B: y = 0.45(400) + 50 = 180 + 50 = $230. Step 6: Compare: Company A costs $200, Company B costs $230. The difference is $230 - $200 = $30. Company A is cheaper by $30. The answer is Company A is cheaper by $30.

7. (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