Problem 21 - Calculus with Concepts in Calculus

Question

Determine the slope \(m\) and \(y\) intercept \(b\) of the line with the given equation. Then sketch the line. \(y=-x\)

Step-by-Step Solution

Answer

The slope \(m\) is \(-1\), and the y-intercept \(b\) is 0. The line passes through the origin with a slope of \(-1\).

Step 1: Identify the form of the equation

The given equation is \( y = -x \). This equation is in the slope-intercept form, \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Determine the slope

From the equation \( y = -x \), compare with \( y = mx + b \). The coefficient of \(x\) is \(-1\). Therefore, the slope \(m\) is \(-1\).

Step 3: Determine the y-intercept

In the equation \( y = -x \), there is no constant term added or subtracted. This means that \(b = 0\). Therefore, the y-intercept is 0.

Step 4: Sketch the line

To sketch the line based on \( y = -x \), start at the y-intercept \((0,0)\). Since the slope \(m\) is \(-1\), for every unit you move to the right (along the x-axis), move down one unit (along the y-axis) to find another point. For example, from \((0,0)\), go to \((1,-1)\) or \((-1,1)\). Draw a straight line through these points.

Key Concepts

Slope

The slope of a line is a number that describes the direction and the steepness of the line. In the slope-intercept form of a linear equation, which is expressed as \( y = mx + b \), the slope is represented by \( m \). This value indicates how much \( y \) changes for a change in \( x \). In simpler terms, it's how slanted the line is. There are some important characteristics to remember about slopes:

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends from left to right.
• A zero slope signifies a horizontal line, which is flat.
• An undefined slope indicates a vertical line, going straight up and down.

In the equation \( y = -x \), the slope \( m \) is \(-1\). This means for every step to the right across the x-axis, the line drops one unit down along the y-axis. This results in a line descending from left to right at a 45-degree angle.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It is an essential component of the slope-intercept form equation \( y = mx + b \), specifically denoted by \( b \). This value tells you where the line will intersect the y-axis, which occurs when \( x = 0 \). Understanding y-intercepts helps to quickly identify one point through which the linear equation passes.

• When \( b = 0 \), the line passes through the origin (0,0).
• If \( b > 0 \), the line intersects the y-axis above the origin.
• If \( b < 0 \), the line intersects the y-axis below the origin.

In our given equation \( y = -x \), the y-intercept \( b \) is 0. This means that the line passes directly through the origin. As there is no constant term added or subtracted in the equation, \( b \) remains 0.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and drawing a straight line that passes through them. Understanding both the slope and the y-intercept aids significantly in sketching an accurate graph.To graph the equation \( y = -x \), follow these steps:

• Begin at the y-intercept, which is the point (0,0) for this equation.
• Use the slope to find additional points. Since the slope is \(-1\), start from the y-intercept and move one unit right and one unit down to get to another point, like (1,-1).
• Alternatively, from (0,0), you can move one unit left and one unit up to reach the point (-1,1).
• Draw a straight line that extends through all these points to complete the graph.

This line depicts the linear relationship represented by the equation \( y = -x \), and understanding the slope and y-intercept makes the graphing process straightforward and efficient.