Graph the line.
$y=-\frac{1}{4} x-4$
Answer (1)
Identify the slope and y-intercept: m = − 4 1 and b = − 4 .
Plot the y-intercept: ( 0 , − 4 ) .
Use the slope to find another point: ( 4 , − 5 ) .
Draw a line through the two points: The graph of y = − 4 1 x − 4 .
Explanation
Understanding the Equation We are asked to graph the line given by the equation y = − 4 1 x − 4 . This is a linear equation in slope-intercept form, y = m x + b , where m represents the slope and b represents the y-intercept.
Identifying Slope and Y-intercept In our equation, y = − 4 1 x − 4 , we can identify the slope m = − 4 1 and the y-intercept b = − 4 . The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0 . So, the y-intercept is the point ( 0 , − 4 ) .
Finding a Second Point To graph the line, we need at least two points. We already have one point, the y-intercept ( 0 , − 4 ) . We can use the slope to find another point. The slope is − 4 1 , which means for every 4 units we move to the right on the x-axis, we move 1 unit down on the y-axis.
Calculating the Second Point Starting from the y-intercept ( 0 , − 4 ) , we move 4 units to the right and 1 unit down. This gives us the point ( 0 + 4 , − 4 − 1 ) = ( 4 , − 5 ) .
Graphing the Line Now we have two points: ( 0 , − 4 ) and ( 4 , − 5 ) . We can plot these points on a coordinate plane and draw a straight line through them. This line represents the graph of the equation y = − 4 1 x − 4 .
Final Answer The line passes through the points ( 0 , − 4 ) and ( 4 , − 5 ) . The slope of the line is − 4 1 , and the y-intercept is -4.
Examples
Understanding linear equations and their graphs is crucial in many real-world applications. For example, if you are tracking the depreciation of an asset over time, you might use a linear equation to model the decrease in value. The slope would represent the rate of depreciation, and the y-intercept would represent the initial value of the asset. Graphing the line helps visualize the asset's value at any point in time.
