Sketch Function Graphs

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Answer Key & Explanations

Sketch Function Graphs · Grade 8 · Worksheet 3

1. Emma is planning a road trip and uses the equation d = 55t to calculate the distance she will travel, where d is distance in miles and t is time in hours. If she wants to travel 330 miles, how many hours will her trip take? Answer: 6 Solution: Start with the given equation: d = 55t Substitute the known distance: 330 = 55t Divide both sides by 55 to solve for t: 330 ÷ 55 = t Calculate the division: 330 ÷ 55 = 6 Therefore, t = 6 The trip will take 6 hours.
   Full step-by-step solution

   Step 1: Start with the given equation: d = 55t Step 2: Substitute the known distance: 330 = 55t Step 3: Divide both sides by 55 to solve for t: 330 ÷ 55 = t Step 4: Calculate the division: 330 ÷ 55 = 6 Step 5: Therefore, t = 6 The trip will take 6 hours.

2. Emma is planning a road trip and needs to calculate her fuel costs. Her car's fuel efficiency is 8.5 liters per 100 kilometers. The total distance of her trip is 425 kilometers, and the current fuel price is $1.40 per liter. How much will Emma spend on fuel for her entire trip? Answer: 50.58 Solution: Calculate the total fuel needed. Fuel efficiency is 8.5 L per 100 km, so for 425 km: Fuel = (8.5 L / 100 km) * 425 km = (0.085 L/km) * 425 km = 36.125 liters Calculate the total cost.
   Full step-by-step solution

   Step 1: Calculate the total fuel needed. Fuel efficiency is 8.5 L per 100 km, so for 425 km: Fuel = (8.5 L / 100 km) * 425 km = (0.085 L/km) * 425 km = 36.125 liters Step 2: Calculate the total cost. Cost = Fuel * Price per liter = 36.125 L * $1.40/L = $50.575 Step 3: Round to the nearest cent for currency. $50.575 rounds to $50.58 The answer is 50.58.

3. Aroha is filling a rectangular fish tank with water. The tank is 12 feet long, 9 feet wide, and 10 feet deep. Water flows from a hose at a constant rate of 15 cubic feet per minute. After 6 minutes, the hose is turned off, and the tank is left to sit for 2 minutes. Then, a drain is opened that removes water at a constant rate of 10 cubic feet per minute for 4 minutes. Sketch a qualitative graph of the water depth (in feet) versus time (in minutes) from the start of filling to the end of draining. Label the axes and key points. Answer: Graph with depth increasing linearly from 0 to 0.833 ft over 6 min, constant at 0.833 ft for 2 min, then decreasing linearly to 0.463 ft over 4 min. Solution: Calculate the volume of the tank: 12 × 9 × 10 = 1080 cubic feet. During filling, water enters at 15 cubic feet per minute. After 6 minutes, volume added = 15 × 6 = 90 cubic feet.
   Full step-by-step solution

   Step 1: Calculate the volume of the tank: 12 × 9 × 10 = 1080 cubic feet. Step 2: During filling, water enters at 15 cubic feet per minute. After 6 minutes, volume added = 15 × 6 = 90 cubic feet. Step 3: Depth after filling = volume added / (length × width) = 90 / (12 × 9) = 90 / 108 = 0.8333 feet. Step 4: During the 2-minute rest, depth stays constant at 0.8333 feet. Step 5: During draining, water leaves at 10 cubic feet per minute for 4 minutes. Volume removed = 10 × 4 = 40 cubic feet. Step 6: Volume remaining = 90 - 40 = 50 cubic feet. Step 7: Depth after draining = 50 / 108 = 0.4630 feet. Step 8: The graph has three segments: from (0,0) to (6, 0.833) with a positive slope; from (6, 0.833) to (8, 0.833) horizontal; from (8, 0.833) to (12, 0.463) with a negative slope. Axes: x-axis labeled Time (minutes), y-axis labeled Depth (feet).

4. Noah is filling a swimming pool with a hose. The water level rises steadily for the first 30 minutes, then he pauses for 10 minutes to check the filter. After that, he turns the hose back on, but the water level rises more slowly because he reduces the water flow. He continues filling at this slower rate for 20 more minutes, then stops completely. Sketch a qualitative graph of the water level (height) in the pool versus time (in minutes). Answer: A graph with three distinct segments: a steep upward sloping line from (0,0) to (30, high), a horizontal line from (30 to 40), then a less steep upward sloping line from (40 to 60). Solution: Identify the three time intervals. From 0 to 30 minutes, the hose is on at full flow, so the water level rises steadily and quickly—this is a straight line with a steep positive slope.
   Full step-by-step solution

   Step 1: Identify the three time intervals. From 0 to 30 minutes, the hose is on at full flow, so the water level rises steadily and quickly—this is a straight line with a steep positive slope. Step 2: From 30 to 40 minutes, the hose is off, so the water level does not change—this is a horizontal line at the height reached at 30 minutes. Step 3: From 40 to 60 minutes, the hose is back on but at a slower flow, so the water level rises steadily but more slowly—this is a straight line with a positive slope that is less steep than the first segment. The graph should have no vertical jumps because the water level changes continuously. The final water level is higher than at any previous time.

5. A scientist is studying bacterial growth in a lab. The initial population of bacteria is 5,000. The population doubles every 3 hours. The scientist models the population using the exponential function P(t) = 5000 * 2^(t/3), where t is the time in hours. What is the population of bacteria after 12 hours? Write your answer as a plain number. Answer: 80000 Solution: Initial population = 5000 Doubling time = 3 hours Model: P(t) = 5000 * 2^(t/3) t = 12 hours P(12) = 5000 * 2^(12/3) 12/3 = 4 So P(12) = 5000 * 2^4 Calculate 2^4 2^4 = 2 * 2 * 2 * 2 = 16 P(12) = 5000 * 16 5000 * 10 = 50000 5000 * 6 = 30000 Add: 50000 + 30000 = 80000 After 12 hours, the population…
   Full step-by-step solution

   Let's solve this step by step. We are given: Initial population = 5000 Doubling time = 3 hours Model: P(t) = 5000 * 2^(t/3) t = 12 hours --- **Step 1: Write the formula with the given time** P(12) = 5000 * 2^(12/3) --- **Step 2: Simplify the exponent** 12/3 = 4 So P(12) = 5000 * 2^4 --- **Step 3: Calculate 2^4** 2^4 = 2 * 2 * 2 * 2 = 16 --- **Step 4: Multiply** P(12) = 5000 * 16 5000 * 10 = 50000 5000 * 6 = 30000 Add: 50000 + 30000 = 80000 --- **Step 5: Interpret the result** After 12 hours, the population is 80000. --- **Final answer:** 80000

6. Aroha starts at her house, walks away from it at a steady pace for 3 minutes, stops to tie her shoe for 1 minute, then walks back home at a faster steady pace for 2 minutes. Sketch a graph of distance from home (in meters) versus time (in minutes). Answer: A qualitative graph with three segments: an upward sloping straight line from (0,0) to (3, d1), a horizontal line from (3, d1) to (4, d1), and a steeper downward sloping straight line from (4, d1) to (6, 0). Solution: Identify the three phases of the story. - Phase 1 (0 to 3 minutes): Aroha walks away from home at a steady pace. Distance increases linearly with time.
   Full step-by-step solution

   Step 1: Identify the three phases of the story. - Phase 1 (0 to 3 minutes): Aroha walks away from home at a steady pace. Distance increases linearly with time. The slope is positive and constant. - Phase 2 (3 to 4 minutes): She stops to tie her shoe. Distance remains constant. The graph is a horizontal line. - Phase 3 (4 to 6 minutes): She walks back home at a faster pace. Distance decreases linearly to zero. The slope is negative and steeper than the first segment because she walks faster. Step 2: Draw the axes. Label the horizontal axis 'Time (minutes)' from 0 to 6. Label the vertical axis 'Distance from home (meters)' from 0 to some maximum (say d1). Step 3: Plot the segments accordingly. The graph starts at the origin (0,0). It rises to a point at (3, d1), then stays horizontal until (4, d1), then falls back to (6, 0). The final segment is steeper than the first. The answer is a qualitative graph with these three segments.

7. 5² + (12 ÷ 4) × 3 - 7 = ? Answer: 27 Solution: Solve inside the parentheses: 12 ÷ 4 = 3 Calculate the exponent: 5² = 25 Perform multiplication: 3 × 3 = 9 Now the expression is: 25 + 9 - 7 Perform addition: 25 + 9 = 34 Perform subtraction: 34 - 7 = 27 The answer is 27.
   Full step-by-step solution

   Step 1: Solve inside the parentheses: 12 ÷ 4 = 3 Step 2: Calculate the exponent: 5² = 25 Step 3: Perform multiplication: 3 × 3 = 9 Step 4: Now the expression is: 25 + 9 - 7 Step 5: Perform addition: 25 + 9 = 34 Step 6: Perform subtraction: 34 - 7 = 27 The answer is 27.