This article provides a comprehensive explanation of the linear equation y = 4x + 8 and its graphical representation. We'll explore how to plot this equation on a Cartesian coordinate system, analyze its key features (slope and y-intercept), and understand its applications in real-world scenarios. Understanding linear equations is fundamental to algebra and has wide-ranging applications in various fields, from physics and engineering to economics and finance.
1. Identifying the Form of the Equation
The equation y = 4x + 8 is a linear equation written in slope-intercept form (y = mx + b), where:
m represents the slope of the line, indicating its steepness or inclination. In this equation, m = 4. A positive slope signifies that the line ascends from left to right.
b represents the y-intercept, which is the point where the line intersects the y-axis (where x = 0). In this case, b = 8. This means the line crosses the y-axis at the point (0, 8).
This form provides a straightforward way to quickly identify the key characteristics of the line and plot it on a graph.
2. Plotting the Line on a Cartesian Coordinate System
To plot the line y = 4x + 8, we can use two methods:
Method 1: Using the slope and y-intercept:
1. Plot the y-intercept: Begin by plotting the point (0, 8) on the y-axis.
2. Use the slope to find another point: The slope of 4 (or 4/1) means that for every 1 unit increase in x, y increases by 4 units. Starting from (0, 8), move 1 unit to the right along the x-axis and 4 units up along the y-axis. This brings you to the point (1, 12).
3. Draw the line: Draw a straight line through the points (0, 8) and (1, 12). This line represents the equation y = 4x + 8.
Method 2: Using two points:
1. Find the y-intercept: When x = 0, y = 4(0) + 8 = 8. This gives us the point (0, 8).
2. Choose another x-value: Let's choose x = 2. Then y = 4(2) + 8 = 16. This gives us the point (2, 16).
3. Plot and connect: Plot the points (0, 8) and (2, 16) and draw a straight line through them. This line, again, represents the equation y = 4x + 8.
3. Interpreting the Slope and Y-intercept
The slope of 4 indicates a steep positive incline. For every unit increase in the x-value (e.g., representing time, distance, or quantity), the y-value (e.g., representing cost, speed, or profit) increases by 4 units.
The y-intercept of 8 represents the initial value or starting point. In a real-world scenario, this could represent a fixed cost, an initial investment, or a base value.
4. Real-world Applications
The equation y = 4x + 8 can model various real-world situations. For example:
Taxi fare: Imagine a taxi service that charges $8 as a base fare and $4 per mile. Here, x represents the number of miles traveled, and y represents the total fare. The equation accurately reflects the cost structure.
Linear growth: The equation could model the growth of a population where the initial population is 8 and increases by 4 units per time period.
Production cost: A company's production cost might be represented by this equation, where x represents the number of units produced, and y represents the total cost. The 8 could represent fixed overhead costs, and the 4x represents the variable costs per unit.
5. Extending the Understanding
This simple linear equation forms the basis for understanding more complex mathematical concepts. It can be used to solve systems of equations, find intersections with other lines, and forms the foundation for understanding linear programming and regression analysis.
Summary
The graph of y = 4x + 8 is a straight line with a slope of 4 and a y-intercept of 8. Understanding its slope and y-intercept allows us to easily plot the line and interpret its meaning in various contexts. This equation provides a fundamental model for representing linear relationships in numerous real-world applications.
FAQs
1. What is the x-intercept of the line y = 4x + 8? To find the x-intercept, set y = 0 and solve for x: 0 = 4x + 8; x = -2. The x-intercept is (-2, 0).
2. How can I determine if a point lies on the line y = 4x + 8? Substitute the x and y coordinates of the point into the equation. If the equation holds true, the point lies on the line.
3. What is the difference between the slope and the y-intercept? The slope represents the rate of change of y with respect to x, while the y-intercept represents the value of y when x is zero.
4. Can this equation be used to model non-linear relationships? No, this equation models linear relationships only. Non-linear relationships require different equations.
5. How would the graph change if the equation were y = -4x + 8? The slope would become negative, resulting in a line descending from left to right, while the y-intercept would remain at 8.
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