Real world examples of proportional and non-proportional relationships

📚 Quick Study Guide

🔢 Proportional Relationships: A relationship between two variables in which their ratio is constant. Represented by the equation $y = kx$, where $k$ is the constant of proportionality. The graph is a straight line through the origin. 📈 Identifying Proportional Relationships: Check if the ratio $\frac{y}{x}$ is the same for all pairs of $x$ and $y$. 🍕 Real-World Example: The cost of pizza slices if each slice costs the same amount. If one slice costs $2, two slices cost $4, and so on. 🚌 Non-Proportional Relationships: A relationship where the ratio between two variables is not constant. The equation is typically in the form $y = mx + b$, where $b \neq 0$. The graph is a straight line that does NOT pass through the origin. ⛽ Real-World Example: The cost of a taxi ride with an initial fee plus a per-mile charge. Even if you travel zero miles, you still pay the initial fee. 📊 Key Difference: Proportional relationships always start at (0,0) on a graph, while non-proportional relationships do not.

🧪 Practice Quiz

1. Which of the following is an example of a proportional relationship?
   1. A) The height of a plant over time, given an initial height.
   2. B) The distance traveled by a car moving at a constant speed.
   3. C) The temperature of water as it's being heated, starting from room temperature.
   4. D) The population of a city that grows by a fixed number of people each year.
2. If $y$ is proportional to $x$, and $y = 12$ when $x = 3$, what is the value of $y$ when $x = 5$?
   1. A) 15
   2. B) 20
   3. C) 24
   4. D) 30
3. Which equation represents a non-proportional relationship?
   1. A) $y = 5x$
   2. B) $y = -2x$
   3. C) $y = x + 3$
   4. D) $y = \frac{1}{2}x$
4. The cost of renting a bicycle is $5 plus $2 per hour. Which type of relationship does this represent?
   1. A) Proportional
   2. B) Non-proportional
   3. C) Both proportional and non-proportional
   4. D) Neither proportional nor non-proportional
5. A recipe calls for 2 cups of flour for every 1 cup of sugar. Is the relationship between the amount of flour and sugar proportional or non-proportional?
   1. A) Proportional
   2. B) Non-proportional
   3. C) Cannot be determined
   4. D) Neither
6. Which graph represents a proportional relationship?
   1. A) A straight line that intersects the y-axis at (0, 2).
   2. B) A curve that increases exponentially.
   3. C) A straight line that passes through the origin (0, 0).
   4. D) A horizontal line.
7. A taxi charges a flat fee of $3 and $0.50 per mile. If $x$ represents the number of miles and $y$ represents the total cost, which equation models this relationship?
   1. A) $y = 0.50x$
   2. B) $y = 3x$
   3. C) $y = 3 + 0.50x$
   4. D) $y = 3 - 0.50x$