Problem 17 - Brief 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

The slope is \(-\frac{1}{2}\) and the y-intercept is (0,0).

Step 1: Identify the Equation Form

The equation given is \( y = -\frac{1}{2} x \). This is in the form of a linear equation \( y = mx + b \). Here, \( m \) represents the slope and \( b \) is the y-intercept.

Step 2: Determine the Slope

From the equation \( y = -\frac{1}{2} x \), the slope \( m \) is \(-\frac{1}{2}\). This means for every unit increase in \( x \), \( y \) decreases by \( \frac{1}{2} \) units.

Step 3: Determine the Y-Intercept

In the equation \( y = -\frac{1}{2}x \), there is no constant term added to \( x \). Therefore, the y-intercept \( b \) is \( 0 \), meaning the line passes through the origin (0,0).

Step 4: Draw the Graph

To draw the graph, start at the y-intercept (0,0). Use the slope \( -\frac{1}{2} \) to determine the next points. From (0,0), move 2 units right along the \( x \)-axis (since our denominator is 2) and 1 unit down (because \( m \) is negative) to reach the point (2,-1). Continue this pattern to sketch the line.

Key Concepts

Slope of a Line

The slope of a line, denoted by \( m \), tells us how steep the line is. It essentially describes how the change in the horizontal direction (x-axis) affects the change in the vertical direction (y-axis). In our equation \( y = -\frac{1}{2} x \), the slope \( m \) is \(-\frac{1}{2}\). This negative slope indicates that as you move from left to right along the line, the line "falls," or goes down.

• A slope of \(-\frac{1}{2}\) means that for every 2 units you move right on the x-axis, you will move 1 unit down on the y-axis.
• If the slope were positive, the line would "rise" or go upwards as you move right.

Understanding slope helps in predicting and sketching the direction and steepness of the line on a graph. It shows the relationship between the variables, representing the rate of change.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis, and it's found when \( x = 0 \). For a linear equation in the form \( y = mx + b \), \( b \) is the y-intercept. In the case of \( y = -\frac{1}{2} x \), since there is no additional constant term, our y-intercept, \( b \), is 0. This means the line passes through the origin, point (0,0).

• The y-intercept gives us a starting point for graphing because it's a specific, fixed location on the graph.
• In many practical situations, the y-intercept can represent starting point data, such as a base cost or initial condition in a problem.

The y-intercept is thus crucial in graphing since it provides one of the key anchor points along the y-axis to draw the line.

Graphing Lines

Graphing lines involves using the slope and y-intercept to draw the path of a linear equation on a coordinate plane. Here are the steps for graphing the line given by the equation \( y = -\frac{1}{2}x \):

• Start by plotting the y-intercept. For this equation, this is the origin: (0,0).
• Next, use the slope to find another point on the line. With a slope of \(-\frac{1}{2}\), you can move 2 units right (positive direction on the x-axis) and 1 unit down (negative direction on the y-axis) to get the next point: (2,-1).
• Continue plotting points using the slope. Additional points might be (4,-2), further helping to maintain accuracy in your graph.

Drawing through these points connects them to form a straight line extending in both directions. The graph represents all potential solutions (x,y) pairs that satisfy the equation. Remember, accurate graphing involves maintaining this consistent slope through all plotted points.