Test Your Knowledge: Linear Functions in Everyday Life (8th Grade)

📚 Quick Study Guide

🔢 What is a Linear Function? A linear function is a relationship between two variables (usually $x$ and $y$) that can be represented by a straight line on a graph. The general form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 📈 Slope ($m$): Represents the rate of change of the line. It's calculated as the change in $y$ divided by the change in $x$: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$. A positive slope means the line goes up from left to right, and a negative slope means it goes down. 📍 Y-Intercept ($b$): The point where the line crosses the y-axis. It's the value of $y$ when $x = 0$. 📝 Graphing Linear Functions: To graph a linear function, you need at least two points. You can find these points by plugging in values for $x$ and solving for $y$, or by using the slope and y-intercept. 🛒 Real-World Examples: Think about the cost of buying candy bars (each bar has a fixed price, that's the slope!) or the distance you travel at a constant speed (speed is the slope!). 💡 Slope-Intercept Form: This is the most common way to write linear equations: $y = mx + b$ 🧭 Finding the Equation: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, first find the slope $m$. Then, use one of the points and the slope to solve for $b$ in the equation $y = mx + b$.

Practice Quiz

1. What does the 'm' represent in the linear equation $y = mx + b$?
   1. A) The y-intercept
   2. B) The slope
   3. C) The x-intercept
   4. D) The area under the line
2. A taxi charges an initial fee of $3 plus $2 per mile. Which equation represents the total cost (y) for x miles?
   1. A) $y = 3x + 2$
   2. B) $y = 2x + 3$
   3. C) $y = 5x$
   4. D) $y = x + 5$
3. What is the y-intercept of the line $y = -4x + 7$?
   1. A) -4
   2. B) 7
   3. C) 4
   4. D) -7
4. Which of the following is NOT a linear function?
   1. A) $y = 2x + 5$
   2. B) $y = x$
   3. C) $y = x^2 + 1$
   4. D) $y = -3x - 7$
5. The line passes through points (0, 2) and (1, 5). What is the slope of the line?
   1. A) 2
   2. B) 3
   3. C) 5
   4. D) -3
6. A lemonade stand sells lemonade for $1.50 per cup. Which graph shows the relationship between cups sold (x) and total earnings (y)?
   1. A) A horizontal line
   2. B) A vertical line
   3. C) A line passing through (0,0) with a positive slope
   4. D) A line passing through (0,0) with a negative slope
7. Which equation represents a line with a slope of -2 and a y-intercept of 5?
   1. A) $y = 5x - 2$
   2. B) $y = -2x + 5$
   3. C) $y = 2x - 5$
   4. D) $y = -5x + 2$