To tackle the problem of sketching the graph of y = x + 4 - 2x - 6, we first need to simplify the equation. By combining like terms, we get y = -x - 2. This equation represents a linear function, which is a straight line when graphed on a coordinate plane. The equation is in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope (m) is -1, and the y-intercept (b) is -2.
### Understanding the Slope and Y-Intercept
The slope of -1 indicates that for every unit increase in x, there is a corresponding decrease in y by 1 unit. This means that the line slopes downward from left to right. The y-intercept of -2 means that the line crosses the y-axis at the point (0, -2). To sketch the graph, we can start by plotting the y-intercept point (0, -2) on the coordinate plane.
### Identifying the X-Intercept
To find the x-intercept, we set y to 0 and solve for x. Substituting y = 0 into the equation y = -x - 2 gives us 0 = -x - 2. Solving for x, we get x = -2. Therefore, the x-intercept is at the point (-2, 0).
### Graphing the Line
With the y-intercept at (0, -2) and the x-intercept at (-2, 0), and knowing the line has a slope of -1, we can sketch the line. Starting from the y-intercept (0, -2), we can move 1 unit to the right (increasing x by 1) and 1 unit down (decreasing y by 1) to get another point on the line. Conversely, moving 1 unit to the left (decreasing x by 1) from the y-intercept would mean going 1 unit up (increasing y by 1). This gives a clear picture of the line's direction.
### Asymptotes
For linear equations like y = -x - 2, there are no asymptotes because the line extends infinitely in both directions without any vertical or horizontal asymptotes.
### Conclusion
Sketching the graph of y = -x - 2 involves plotting the y-intercept at (0, -2) and the x-intercept at (-2, 0), and then drawing a line through these points with a slope of -1. The line represents all possible solutions to the equation y = -x - 2 and extends infinitely in both directions with no asymptotes.
The graph of y = x + 4 - 2x - 6, simplified to y = -x - 2, is a straight line. By understanding the slope and intercepts, one can easily visualize or draw the graph of this linear equation on a coordinate plane.
Key phrases: graphing linear equations, linear functions, slope intercept form, finding x and y intercepts, sketching lines on a coordinate plane, equation of a line, linear equation graphing, slope and y intercept.
