Problem 17 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ y=-\frac{1}{2} x $$

Step-by-Step Solution

Answer

Slope \(m = -\frac{1}{2}\) and \(y\)-intercept is \((0, 0)\).

Step 1: Identify the Form of the Equation

The equation given is written in slope-intercept form, which is \(y = mx + b\). In this form, \(m\) represents the slope and \(b\) represents the \(y\)-intercept.

Step 2: Determine the Slope

In this equation, \(y = -\frac{1}{2}x\), the coefficient of \(x\) is \(-\frac{1}{2}\). Therefore, the slope \(m = -\frac{1}{2}\).

Step 3: Identify the y-intercept

Since the equation is \(y = -\frac{1}{2}x + 0\), the \(y\)-intercept \(b = 0\). This means the \(y\)-intercept is at the point \((0, 0)\).

Step 4: Graph the Equation

To draw the graph, start at the point \((0,0)\) which is the \(y\)-intercept. From there, use the slope \(-\frac{1}{2}\), which means for every 1 unit you move right along the x-axis, move 1/2 unit down along the y-axis, or for every 2 units you move right, move 1 unit down. Draw the line through these points.

Key Concepts

Slope-Intercept Form

One of the essential forms to express linear equations is the slope-intercept form. This equation is typically written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the \(y\)-intercept. This form makes it simple to identify these two critical components of the line.

• Slope (\(m\)): This describes how steep the line is. It tells you how much \(y\) changes for a unit change in \(x\).
• Y-Intercept (\(b\)): This is the point where your line crosses the \(y\)-axis (when \(x = 0\)).

The beauty of the slope-intercept form lies in its simplicity in graphing linear equations and understanding the geometric interpretation of the line's slope and starting point on the \(y\)-axis.

Graphing Linear Equations

Once you have an equation in slope-intercept form \(y = mx + b\), graphing becomes straightforward. Follow these steps to plot your linear equation:

• Start with the y-intercept (\(b\)): Look at your equation to identify \(b\), the starting point on the \(y\)-axis. This is the point \((0, b)\).
• Use the slope (\(m\)): The slope tells you the rise over run. For instance, if \(m = -\frac{1}{2}\), each time you move 1 unit to the right, move \(1/2\) unit down.

Connect the points with a straight line and extend it in both directions. This results in a graph representing all the solutions to the equation. The clarity of this method makes it a favorite among learners who prefer visual understanding.

Y-Intercept

Understanding the \(y\)-intercept \((b)\) is crucial as it serves as a starting point for graphing your line. The \(y\)-intercept is where the line crosses the \(y\)-axis, meaning at this point \(x = 0\).

• In our example, the equation is \(y = -\frac{1}{2}x + 0\). Here, the \(y\)-intercept \(b\) is \(0\), placing the initial point at \((0, 0)\).
• When graphing, always locate the \(y\)-intercept first. It is your anchor point starting on the \(y\)-axis.

This point enables easy graphing along with the slope, allowing you to visually create a line with accurate representation of the equation on a coordinate plane. Prioritizing the \(y\)-intercept helps set up a clear foundation from which to use the provided slope.