Graph the equation $y=-\frac{1}{4} x-3$ by plotting points.
Answer (1)
Identify the slope and y-intercept from the equation: m = − 4 1 and b = − 3 .
Determine the y-intercept point: (0, -3).
Use the slope to find another point on the line: (4, -4).
Draw a straight line through the two points (0, -3) and (4, -4).
Explanation
Understanding the Equation We are given the equation of a line: y = − 4 1 x − 3 . Our goal is to graph this equation without plotting individual points, but rather by understanding the properties of the line based on its equation.
Identifying Slope and Y-Intercept The equation is in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y-intercept. In our case, m = − 4 1 and b = − 3 .
Finding the Y-Intercept The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0 . In our equation, the y-intercept is -3, so the line passes through the point (0, -3).
Using the Slope to Find Another Point The slope of the line is − 4 1 . This means that for every 4 units we move to the right along the x-axis, the line moves 1 unit down along the y-axis. We can use this information to find another point on the line. Starting from the y-intercept (0, -3), if we move 4 units to the right (to x = 4 ), we move 1 unit down (to y = − 4 ). So, another point on the line is (4, -4).
Graphing the Line Now we have two points on the line: (0, -3) and (4, -4). We can draw a straight line through these two points to represent the graph of the equation y = − 4 1 x − 3 .
Final Answer The graph is a straight line that passes through the points (0, -3) and (4, -4). The line has a negative slope, so it goes downwards as we move from left to right.
Examples
Linear equations are used in many real-world scenarios. For example, if you are saving money at a constant rate, the equation representing your savings over time is a linear equation. Similarly, if you are traveling at a constant speed, the equation representing the distance you have traveled over time is also a linear equation. Understanding how to graph linear equations helps you visualize these relationships and make predictions about future outcomes.
