Compare Functions

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Answer Key & Explanations

Compare Functions · Grade 8 · Worksheet 3

1. Function A: y = 8x + 5. Function B: table below. Which function has the greater y-intercept? x | y 0 | 12 2 | 28 4 | 44 6 | 60 Answer: Function B Solution: Find the y-intercept of Function A. In the equation y = 8x + 5, the y-intercept is the constant term, which is 5. 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 = 8x + 5, the y-intercept is the constant term, which is 5. Step 2: Find the y-intercept of Function B. From the table, when x = 0, y = 12. So the y-intercept is 12. Step 3: Compare the y-intercepts. Function A has y-intercept 5, Function B has y-intercept 12. Since 12 > 5, Function B has the greater y-intercept. The answer is Function B.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (12,5). A second right triangle is drawn with vertices at (0,0), (18,0), and (18,7.5). Are these triangles similar? If so, what is the scale factor from the smaller triangle to the larger triangle? Answer: 1.5 Solution: Check if the triangles are similar by comparing ratios of corresponding sides. For the smaller triangle, the vertical side is 5 units and horizontal side is 12 units.
   Full step-by-step solution

   Step 1: Check if the triangles are similar by comparing ratios of corresponding sides. Step 2: For the smaller triangle, the vertical side is 5 units and horizontal side is 12 units. Step 3: For the larger triangle, the vertical side is 7.5 units and horizontal side is 18 units. Step 4: Compare vertical sides: 7.5 ÷ 5 = 1.5 Step 5: Compare horizontal sides: 18 ÷ 12 = 1.5 Step 6: Since both ratios equal 1.5, the triangles are similar. Step 7: The scale factor from smaller to larger triangle is 1.5.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (0,6). A circle is drawn such that its diameter is the hypotenuse of the triangle. What is the area of the circle? (Use π = 3.14) Answer: 78.5 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 8 units (from (0,0) to (8,0)) and 6 units (from (0,0) to (0,6)).
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 8 units (from (0,0) to (8,0)) and 6 units (from (0,0) to (0,6)). Hypotenuse = sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 units. Step 2: The hypotenuse is the diameter of the circle. Diameter = 10 units Radius = Diameter/2 = 10/2 = 5 units Step 3: Calculate the area of the circle. Area = π × radius^2 = 3.14 × 5^2 = 3.14 × 25 = 78.5 square units. The answer is 78.5.

4. Liam is comparing two different phone plans. Plan A costs $20 per month plus $0.10 per gigabyte of data used. Plan B has a flat fee of $35 per month for unlimited data. Liam wants to determine how many gigabytes he would need to use for both plans to cost the same amount. Write an equation to represent this situation and solve for the number of gigabytes. Answer: 150 Solution: Plan A costs $20 per month plus $0.10 per gigabyte. So cost for Plan A = 20 + 0.10 × g Plan B costs a flat $35 per month.
   Full step-by-step solution

   Let's define the number of gigabytes used as g. Plan A costs $20 per month plus $0.10 per gigabyte. So cost for Plan A = 20 + 0.10 × g Plan B costs a flat $35 per month. So cost for Plan B = 35 We want the number of gigabytes where both plans cost the same: 20 + 0.10 × g = 35 Step 1: Subtract 20 from both sides to isolate the term with g. 0.10 × g = 35 − 20 0.10 × g = 15 Step 2: Divide both sides by 0.10 to solve for g. g = 15 ÷ 0.10 Step 3: Calculate the division. 15 ÷ 0.10 = 15 ÷ (1/10) = 15 × 10 = 150 So, g = 150 gigabytes. This means Liam would need to use 150 gigabytes for both plans to cost the same amount.

5. 2³ × (6² - 4²) = ? Answer: 160 Solution: Calculate the exponents first: 2³ = 8 and 6² = 36 and 4² = 16 Perform the subtraction inside the parentheses: 36 - 16 = 20 Multiply the results: 8 × 20 = 160 The answer is 160.
   Full step-by-step solution

   Step 1: Calculate the exponents first: 2³ = 8 and 6² = 36 and 4² = 16 Step 2: Perform the subtraction inside the parentheses: 36 - 16 = 20 Step 3: Multiply the results: 8 × 20 = 160 The answer is 160.

6. 2³ × (4² - 3²) = ? Answer: 56 Solution: 2³ × (4² - 3²) = ? 4² = 4 × 4 = 16 3² = 3 × 3 = 9 4² - 3² = 16 - 9 = 7 2³ × 7 Evaluate 2³ 2³ = 2 × 2 × 2 = 8 8 × 7 8 × 7 = 56 Final Answer: 56
   Full step-by-step solution

   Let's solve the problem step by step. We have: 2³ × (4² - 3²) = ? --- **Step 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 --- **Step 2: Evaluate 2³** 2³ = 2 × 2 × 2 = 8 So now we have: 8 × 7 --- **Step 3: Multiply** 8 × 7 = 56 --- **Final Answer:** 56

7. Function A: y = 7x + 9. Function B: table below. Which function has the greater y-intercept? x | y 1 | 15 3 | 23 5 | 31 7 | 39 Answer: Function A Solution: Find the y-intercept of Function A. In y = 7x + 9, the y-intercept is the constant term, which is 9. Find the slope of Function B using two points, such as (1, 15) and (3, 23).
   Full step-by-step solution

   Step 1: Find the y-intercept of Function A. In y = 7x + 9, the y-intercept is the constant term, which is 9. Step 2: Find the slope of Function B using two points, such as (1, 15) and (3, 23). Slope = (23 - 15) / (3 - 1) = 8 / 2 = 4. Step 3: Use the slope and one point to find the y-intercept of Function B. Use y = mx + b with point (1, 15) and slope m = 4: 15 = 4(1) + b → 15 = 4 + b → b = 11. Step 4: Compare the y-intercepts. Function A has y-intercept 9, Function B has y-intercept 11. Since 9 < 11, Function A has the smaller y-intercept. The question asks which has the greater y-intercept, so Function B has the greater y-intercept. The answer is Function B.

8. Liam is comparing two internet plans. Plan A costs $15 per month plus $0.50 per gigabyte used. Plan B has a flat fee of $35 per month for unlimited data. Liam creates a table showing the total monthly cost for Plan A at different usage levels: 10GB costs $20, 20GB costs $25, and 30GB costs $30. He also graphs both plans, with Plan B appearing as a horizontal line. At what data usage do the two plans cost the same amount? Answer: 40 gigabytes Solution: Write the cost equations for both plans. Cost = 15 + 0.50 × (number of gigabytes) Let \( x \) = gigabytes used. Cost_A = 15 + 0.50x Cost_B = 35 (flat fee) Set the costs equal to find when they are the same.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Write the cost equations for both plans.** Plan A: Cost = 15 + 0.50 × (number of gigabytes) Let \( x \) = gigabytes used. So: Cost_A = 15 + 0.50x Plan B: Cost_B = 35 (flat fee) --- **Step 2: Set the costs equal to find when they are the same.** 15 + 0.50x = 35 --- **Step 3: Solve for x.** Subtract 15 from both sides: 0.50x = 35 - 15 0.50x = 20 Divide both sides by 0.50: x = 20 / 0.50 x = 40 --- **Step 4: Interpret the result.** At 40 gigabytes of data usage, Plan A costs the same as Plan B ($35). --- **Final answer:** 40 gigabytes