Compare Functions

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Answer Key & Explanations

Compare Functions · Grade 8 · Worksheet 1

1. Function A: y = 13x + 9. Function B: table below. Which function has the greater y-intercept? x | y 0 | 21 2 | 47 4 | 73 6 | 99 Answer: Function B Solution: Find the y-intercept of Function A. In the equation y = 13x + 9, the y-intercept is the constant term, which is 9. 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 + 9, the y-intercept is the constant term, which is 9. Step 2: Find the y-intercept of Function B. The table shows that when x = 0, y = 21. So the y-intercept of Function B is 21. Step 3: Compare the y-intercepts. Function A has y-intercept 9, Function B has y-intercept 21. Since 21 > 9, Function B has the greater y-intercept. The answer is Function B.

2. (3 × 10⁴) × (4 × 10⁻²) = ? Answer: 1200 Solution: Multiply the coefficients: 3 × 4 = 12 Add the exponents of 10: 4 + (-2) = 2 Combine the results: 12 × 10² Calculate 12 × 100 = 1200 The answer is 1200.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 3 × 4 = 12 Step 2: Add the exponents of 10: 4 + (-2) = 2 Step 3: Combine the results: 12 × 10² Step 4: Calculate 12 × 100 = 1200 The answer is 1200.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by translating the first triangle 4 units to the left and 3 units down, then reflecting it across the x-axis. What are the coordinates of the vertices of the final transformed triangle? Answer: (-4,-3), (-4,-11), (-10,-3) Solution: Geometric transformations on the coordinate plane follow specific rules. Translation moves all points by adding or subtracting from their coordinates.
   Full step-by-step solution

   Geometric transformations on the coordinate plane follow specific rules. Translation moves all points by adding or subtracting from their coordinates. Reflection across the x-axis keeps x-coordinates the same but changes the sign of y-coordinates. When performing multiple transformations, the order matters and each transformation affects all vertices of the shape equally.

4. Emma is comparing two different gym membership options. Gym A charges a $40 monthly fee plus a one-time $50 registration fee. Gym B charges a $30 monthly fee plus a one-time $80 registration fee. Emma wants to know after how many months the total cost of both gym memberships would be equal. How many months would that be? Answer: 3 Solution: Write equations for the total cost of each gym after m months Gym A: Total cost = 50 + 40m Gym B: Total cost = 80 + 30m Set the equations equal to find when costs are the same 50 + 40m = 80 + 30m Subtract 30m from both sides 50 + 10m = 80 Subtract 50 from both sides 10m = 30 Divide both sides by…
   Full step-by-step solution

   Step 1: Write equations for the total cost of each gym after m months Gym A: Total cost = 50 + 40m Gym B: Total cost = 80 + 30m Step 2: Set the equations equal to find when costs are the same 50 + 40m = 80 + 30m Step 3: Subtract 30m from both sides 50 + 10m = 80 Step 4: Subtract 50 from both sides 10m = 30 Step 5: Divide both sides by 10 m = 3 After 3 months, both gym memberships would cost the same total amount.

5. (3 × 10⁴) × (2 × 10⁻²) = ? Answer: 600 Solution: Multiply the coefficients: 3 × 2 = 6 Add the exponents: 4 + (-2) = 2 Combine the results: 6 × 10² Calculate 6 × 100 = 600 The answer is 600.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 3 × 2 = 6 Step 2: Add the exponents: 4 + (-2) = 2 Step 3: Combine the results: 6 × 10² Step 4: Calculate 6 × 100 = 600 The answer is 600.

6. Liam is comparing two different streaming services. Service A charges a $5 monthly fee plus $0.50 per movie watched. Service B charges a $12 monthly fee plus $0.25 per movie watched. Liam wants to know how many movies he would need to watch for both services to cost the same amount. Write an equation to represent this situation and solve for the number of movies. Answer: 28 Solution: Service A cost = monthly fee + cost per movie × number of movies Service A cost = 5 + 0.50 × m Service B cost = monthly fee + cost per movie × number of movies Service B cost = 12 + 0.25 × m 5 + 0.50m = 12 + 0.25m Subtract 0.25m from both sides to get m terms on one side.
   Full step-by-step solution

   Let's define the number of movies watched as m. Service A cost = monthly fee + cost per movie × number of movies Service A cost = 5 + 0.50 × m Service B cost = monthly fee + cost per movie × number of movies Service B cost = 12 + 0.25 × m We want the costs to be equal: 5 + 0.50m = 12 + 0.25m Step 1: Subtract 0.25m from both sides to get m terms on one side. 5 + 0.50m - 0.25m = 12 + 0.25m - 0.25m 5 + 0.25m = 12 Step 2: Subtract 5 from both sides to isolate the m term. 5 + 0.25m - 5 = 12 - 5 0.25m = 7 Step 3: Divide both sides by 0.25 to solve for m. m = 7 / 0.25 Step 4: Calculate 7 divided by 0.25. Dividing by 0.25 is the same as multiplying by 4. m = 7 × 4 m = 28 So, Liam would need to watch 28 movies for both services to cost the same.

7. 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.

8. Liam is comparing two different streaming services for his gaming videos. Service A charges a flat fee of $15 per month plus $0.25 per gigabyte of data used. Service B charges a flat fee of $8 per month plus $0.40 per gigabyte of data used. Liam wants to know at what number of gigabytes the total monthly cost would be the same for both services. How many gigabytes would that be? Answer: 46.67 Solution: Service A cost: flat fee $15 plus $0.25 per gigabyte Cost_A = 15 + 0.25x Service B cost: flat fee $8 plus $0.40 per gigabyte Cost_B = 8 + 0.40x We want the number of gigabytes x where both costs are equal: 15 + 0.25x = 8 + 0.40x Subtract 8 from both sides 15 - 8 + 0.25x = 8 - 8 + 0.40x 7 + 0.25x…
   Full step-by-step solution

   Let's define the number of gigabytes used as x. Service A cost: flat fee $15 plus $0.25 per gigabyte Cost_A = 15 + 0.25x Service B cost: flat fee $8 plus $0.40 per gigabyte Cost_B = 8 + 0.40x We want the number of gigabytes x where both costs are equal: 15 + 0.25x = 8 + 0.40x Step 1: Subtract 8 from both sides 15 - 8 + 0.25x = 8 - 8 + 0.40x 7 + 0.25x = 0.40x Step 2: Subtract 0.25x from both sides 7 + 0.25x - 0.25x = 0.40x - 0.25x 7 = 0.15x Step 3: Divide both sides by 0.15 x = 7 / 0.15 Step 4: Calculate 7 divided by 0.15 7 / 0.15 = 700 / 15 700 ÷ 15 = 46.6666... So x ≈ 46.67 gigabytes This means at 46.67 gigabytes of data used, both services cost the same.