Problem 17 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line with slope \(-1\) and \(y\) intercept 0

Step-by-Step Solution

Answer

The line's equation is \(y = -x\).

Step 1: Identify Given Values

We are given that the slope \(m\) of the line is \(-1\) and the \(y\)-intercept \(b\) is 0.

Step 2: Write the Slope-Intercept Form

The equation of a line in slope-intercept form is \(y = mx + b\). Use \(m = -1\) and \(b = 0\) to fill in the equation.

Step 3: Substitute Given Values

Substitute the values of \(m\) and \(b\) into the slope-intercept form: \(y = -1x + 0\).

Step 4: Simplify the Equation

The equation simplifies to \(y = -x\), as adding zero does not change the expression.

Step 5: Sketch the Line

To sketch the line, plot the \(y\)-intercept (0,0) on the graph. Use the slope to determine another point: from (0,0), move down 1 unit and right 1 unit to reach (1,-1). Draw the line through these points.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express the equation of a line using two key components: the slope and the y-intercept. It's written in the format:\[ y = mx + b \]where:

• \(m\) represents the slope of the line. The slope indicates the steepness of the line and the direction it moves. A positive slope means the line rises from left to right, while a negative slope shows it falls from left to right.
• \(b\) is the y-intercept of the line. This is the point where the line crosses the y-axis. It tells you the value of \(y\) when \(x = 0\).

The slope-intercept form is widely used because it provides a straightforward way to graph a line and understand its behavior. By plugging in the values of \(m\) and \(b\), you can quickly sketch or calculate the position of the line on a graph.

Graphing Lines

Graphing lines using the slope-intercept form is intuitive once you understand the basics.To graph a line:

• Start with the y-intercept: Plot the point corresponding to the value of \(b\) on the y-axis. This is your starting point.
• Use the slope: From the y-intercept, use the slope \(m\), which is typically represented as a fraction \(\frac{rise}{run}\). This tells you how to move from the y-intercept: the "rise" indicates how many units to go up or down, and the "run" tells you how many units to go left or right.
• Draw the line: With these points plotted, draw a straight line through them extending it across the graph.

By following these steps, you can efficiently graph any line in slope-intercept form. For example, with the equation \(y = -x\), starting at (0,0), move down 1 unit and to the right 1 unit to get to the next point (1,-1), and then draw the line through these points.

Slope of a Line

The slope of a line is a crucial concept in understanding linear equations. It is defined as the "steepness" of the line and is calculated as the change in y-coordinates divided by the change in x-coordinates between two distinct points on the line. Often expressed as:\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]The slope tells you how much \(y\) changes for a change in \(x\):

• Positive slope: Line rises from left to right.
• Negative slope: Line falls from left to right.
• Zero slope: Line is horizontal.
• Undefined slope: Line is vertical.

Understanding the slope helps in determining the direction and angle at which the line travels. In our example, with a slope of \(-1\), it indicates that for every unit you move to the right, the line moves down by one unit. This negative slope results in a line that descends as it proceeds from left to right on the graph.