- Identify the slope \(m = -4\) and y-intercept \(b = -1\) from the equation \(y = -4x - 1\).
- Plot the y-intercept at \((0, -1)\) and use the slope to find a second point at \((1, -5)\).
- Draw a straight line passing through \((0, -1)\) and \((1, -5)\).
- Verify the graph by checking an additional point such as \((-1, 3)\) and identifying the x-intercept at \((-0.25, 0)\).
- The final graph is a line with a negative slope passing through these points \(\boxed{y = -4x - 1}\).
Explanation
- Analyzing the Equation
The given equation is \(y = -4x - 1\). This is a linear equation expressed in slope-intercept form, which is given by \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. In our specific equation, the slope \(m = -4\) and the y-intercept \(b = -1\). The slope of \(-4\) indicates that for every 1 unit move to the right along the x-axis, the graph moves down 4 units along the y-axis. The y-intercept of \(-1\) means that the line crosses the y-axis at the point \((0, -1)\).
- Steps to Graph the Equation
To draw the graph, first plot the y-intercept at \((0, -1)\) on the coordinate plane. Next, apply the slope \(m = -4 = \frac{-4}{1}\). Starting from the y-intercept \((0, -1)\), move 4 units down and 1 unit to the right to locate a second point. Calculating the coordinates of this point, we move from \(y = -1\) down 4 units to get \(y = -5\), and from \(x = 0\) right 1 unit to get \(x = 1\). Thus, the second point is \((1, -5)\). Finally, draw a straight line that connects these two points: \((0, -1)\) and \((1, -5)\). Extend this line in both directions to show that it continues indefinitely.
- Verifying the Graph
To verify the accuracy of the graph, calculate the coordinates of a third point using a different x-value. For instance, if we set \(x = -1\), the equation becomes \(y = -4(-1) - 1 = 4 - 1 = 3\). The point \((-1, 3)\) should lie on the line you drew. If the line passes through \((-1, 3)\), \((0, -1)\), and \((1, -5)\), the graph is correct. Additionally, you can check the x-intercept by setting \(y = 0\): \(0 = -4x - 1\), which gives \(x = -1/4\). The line should cross the x-axis at \((-0.25, 0)\). These points confirm the slope and position of the line.
Examples
Linear equations like \(y = -4x - 1\) are essential for modeling relationships where one variable changes at a constant rate relative to another. For example, if you are filling a water tank that already has a small leak (represented by the \(-1\)), and you are drawing water from it at a constant speed, the remaining water level over time could be modeled by this linear equation. Understanding how to graph these relationships allows you to quickly visualize how resources deplete or accumulate, helping you make informed decisions about time management and resource allocation.