Sketch Function Graphs

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Answer Key & Explanations

Sketch Function Graphs · Grade 8 · Worksheet 2

1. Liam is designing a rectangular garden with a perimeter of 40 meters. The length of the garden is 4 meters more than its width. What are the dimensions of Liam's garden? Answer: width = 8 meters, length = 12 meters Solution: Let the width of the garden be \( w \) meters. The length is 4 meters more than the width, so length \( l = w + 4 \).
   Full step-by-step solution

   Let's solve this step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 4 meters more than the width, so length \( l = w + 4 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times \text{length} + 2 \times \text{width} \) Given \( P = 40 \), we have: \( 2l + 2w = 40 \) --- **Step 3: Substitute \( l = w + 4 \) into the perimeter equation** \( 2(w + 4) + 2w = 40 \) --- **Step 4: Simplify and solve for \( w \)** \( 2w + 8 + 2w = 40 \) \( 4w + 8 = 40 \) Subtract 8 from both sides: \( 4w = 32 \) Divide by 4: \( w = 8 \) --- **Step 5: Find the length** \( l = w + 4 = 8 + 4 = 12 \) --- **Step 6: Final answer** Width = 8 meters Length = 12 meters

2. Emma is filling a rectangular fish tank with water. The water level rises steadily for 5 minutes, then stays constant for 2 minutes while she adjusts the filter, then rises at a slower steady rate for 3 more minutes until the tank is full. Sketch a graph of water depth (in cm) vs. time (in minutes). Answer: A qualitative graph with three segments: a straight line with positive slope from (0,0) to (5, 20), a horizontal line from (5, 20) to (7, 20), and a straight line with a smaller positive slope from (7, 20) to (10, 30). Solution: The first segment (0 to 5 minutes) shows a steady rise. Draw a straight line with a positive slope starting at the origin (0,0). Step 2: The second segment (5 to 7 minutes) shows no change.
   Full step-by-step solution

   Step 1: The first segment (0 to 5 minutes) shows a steady rise. Draw a straight line with a positive slope starting at the origin (0,0). For example, if the water rises 4 cm per minute, after 5 minutes the depth is 20 cm. Step 2: The second segment (5 to 7 minutes) shows no change. Draw a horizontal line at the depth reached (20 cm) from time 5 to 7 minutes. Step 3: The third segment (7 to 10 minutes) shows a slower steady rise. Draw a straight line with a smaller positive slope from (7, 20) to (10, 30), meaning it rises 10 cm over 3 minutes (about 3.3 cm per minute). The final graph has three connected line segments: steep up, flat, less steep up.

3. Noah is filling a swimming pool. The water level rises quickly at first, then slows down as the pool gets deeper, and finally stays constant when the pool is full. Sketch a graph of water depth vs. time. Answer: A qualitative graph showing a curve that increases steeply at first, then gradually flattens, and finally becomes horizontal. Solution: Identify the axes: horizontal axis is time, vertical axis is water depth. At the start (time = 0), depth is 0, so the graph begins at the origin.
   Full step-by-step solution

   Step 1: Identify the axes: horizontal axis is time, vertical axis is water depth. Step 2: At the start (time = 0), depth is 0, so the graph begins at the origin. Step 3: The water level rises quickly at first, so the graph has a steep positive slope initially. Step 4: As the pool gets deeper, the rate slows, so the slope becomes less steep (the curve bends toward horizontal). Step 5: When the pool is full, the depth stays constant, so the graph becomes a horizontal line (slope = 0). The final sketch is a curve that starts steep, gradually flattens, and ends with a horizontal segment.

4. Emma is organizing a school fundraiser and needs to create a budget. She has $500 to spend on supplies. She buys t-shirts that cost $8 each and water bottles that cost $5 each. If she buys 40 t-shirts, how many water bottles can she buy while spending exactly her $500 budget? Answer: 36 Solution: Step 1: Calculate the cost of the t-shirts: 40 t-shirts × $8 per t-shirt = $320 Step 2: Subtract the t-shirt cost from the total budget: $500 - $320 = $180 remaining Step 3: Divide the remaining money by the cost per water bottle: $180 ÷ $5 per water bottle = 36 water bottles Step 4: Verify: (40…
   Full step-by-step solution

   Step 1: Calculate the cost of the t-shirts: 40 t-shirts × $8 per t-shirt = $320 Step 2: Subtract the t-shirt cost from the total budget: $500 - $320 = $180 remaining Step 3: Divide the remaining money by the cost per water bottle: $180 ÷ $5 per water bottle = 36 water bottles Step 4: Verify: (40 × $8) + (36 × $5) = $320 + $180 = $500 The answer is 36 water bottles.

5. Sophia is filling a cylindrical tank with water at a constant rate. She then stops filling for 6 minutes to check the tank. After that, she continues filling at the same constant rate until the tank is full. Sketch a graph of the water height in the tank versus time. Answer: A graph with three distinct segments: a straight line with positive slope from the origin, a horizontal line segment, and another straight line with the same positive slope as the first segment, ending at a higher height. Solution: Identify the three phases of the scenario: filling, stopping, and filling again. During the first filling phase, water is added at a constant rate, so the height increases steadily over time. This is represented by a straight line with a positive slope starting from the origin (0,0).
   Full step-by-step solution

   Step 1: Identify the three phases of the scenario: filling, stopping, and filling again. Step 2: During the first filling phase, water is added at a constant rate, so the height increases steadily over time. This is represented by a straight line with a positive slope starting from the origin (0,0). Step 3: During the 6-minute stop, no water is added, so the height remains constant. This is represented by a horizontal line segment at the height reached at the end of the first filling phase. Step 4: During the second filling phase, water is again added at the same constant rate, so the height increases at the same rate as before. This is represented by another straight line with the same positive slope as the first segment, starting from the end of the horizontal segment and continuing until the tank is full. Step 5: The final graph has three connected segments: a rising line, a flat line, and another rising line with the same slope as the first. The answer is a sketch showing this pattern.

6. 3² × (4 + 2) ÷ √9 = ? Answer: 18 Solution: Calculate inside the parentheses: (4 + 2) = 6 Calculate the exponent: 3² = 9 Calculate the square root: √9 = 3 Now the expression is: 9 × 6 ÷ 3 Multiply and divide from left to right: 9 × 6 = 54 54 ÷ 3 = 18 The answer is 18.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: (4 + 2) = 6 Step 2: Calculate the exponent: 3² = 9 Step 3: Calculate the square root: √9 = 3 Step 4: Now the expression is: 9 × 6 ÷ 3 Step 5: Multiply and divide from left to right: 9 × 6 = 54 Step 6: 54 ÷ 3 = 18 The answer is 18.

7. Emma is tracking the growth of her sunflower plant. In the first week, it grew 8 cm. Each subsequent week, it grew 75% of the amount it grew the previous week. What is the total height the sunflower will have grown after a very long time (the sum of this infinite geometric series)? Answer: 32 Solution: Identify the first term (a) of the series. The growth in the first week is 8 cm, so a = 8. Identify the common ratio (r).
   Full step-by-step solution

   Step 1: Identify the first term (a) of the series. The growth in the first week is 8 cm, so a = 8. Step 2: Identify the common ratio (r). The plant grows 75% of the previous week's growth each time. 75% is 0.75, so r = 0.75. Step 3: The formula for the sum of an infinite geometric series is S = a / (1 - r), provided |r| < 1. Step 4: Substitute the known values into the formula: S = 8 / (1 - 0.75). Step 5: Calculate the denominator: 1 - 0.75 = 0.25. Step 6: Perform the division: 8 / 0.25 = 32. Step 7: The total growth the sunflower approaches over a very long time is 32 cm.

8. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. If the area of the garden must be 54 square meters, what are the dimensions of the garden? Answer: width = 4.5 m, length = 12 m Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: \text{length} = 2w + 3 Area of a rectangle = length × width Given area = 54 square meters: (2w + 3) \times w = 54 2w^2 + 3w = 54 2w^2 + 3w - 54 = 0 w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} Here \(…
   Full step-by-step solution

   Let's solve the problem step by step. --- **Step 1: Define the variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: \[ \text{length} = 2w + 3 \] --- **Step 2: Write the area equation** Area of a rectangle = length × width Given area = 54 square meters: \[ (2w + 3) \times w = 54 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 54 \] \[ 2w^2 + 3w - 54 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here \( a = 2 \), \( b = 3 \), \( c = -54 \). First, compute the discriminant: \[ b^2 - 4ac = 3^2 - 4(2)(-54) = 9 + 432 = 441 \] \[ \sqrt{441} = 21 \] Now: \[ w = \frac{-3 \pm 21}{2 \times 2} = \frac{-3 \pm 21}{4} \] --- **Step 5: Two possible solutions for \( w \)** First: \[ w = \frac{-3 + 21}{4} = \frac{18}{4} = 4.5 \] Second: \[ w = \frac{-3 - 21}{4} = \frac{-24}{4} = -6 \] --- **Step 6: Interpret the solutions** Width cannot be negative, so \( w = 4.5 \) meters. --- **Step 7: Find the length** \[ \text{length} = 2w + 3 = 2(4.5) + 3 = 9 + 3 = 12 \] --- **Step 8: Verify** Area = length × width = \( 12 \times 4.5 = 54 \) — correct. --- **Final answer:** Width = 4.5 m, Length = 12 m