Problem 77 - Calculus Volume 1

Question

For the following exercises, for each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b,\) if any, and b. graph the line. $$ f(x)=-6 x $$

Step-by-Step Solution

Answer

The slope is -6, and the y-intercept is 0. Graph by plotting (0,0) and using the slope to find (1,-6).

Step 1: Identify the Equation Form

The given equation is \( f(x) = -6x \). This is a linear equation in the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Determine the Slope (m)

In the equation \( f(x) = -6x \), the coefficient of \( x \) is \(-6\). Therefore, the slope \( m = -6 \).

Step 3: Determine the Y-intercept (b)

The given equation \( f(x) = -6x \) can be compared to \( y = mx + b \). Since there is no constant term, the y-intercept \( b \) is \( 0 \).

Step 4: Graph the Line

To graph the line, start at the y-intercept \( (0,0) \) on the coordinate plane. Since the slope \( m = -6 \), this means for every 1 unit increase in \( x \), \( y \) decreases by 6 units. From point \( (0,0) \), move right 1 unit to \( x = 1 \) and go down 6 units to \( (1,-6) \). Draw a straight line through these points to represent the equation.

Key Concepts

Understanding the Slope

The slope of a linear equation is a crucial concept that tells us how steep the line is and the direction it goes. In general, the slope is represented by \( m \) in the equation of a line, \( y = mx + b \). Here's what you need to know about the slope:

• The slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.
• Positive slopes mean the line inclines upward from left to right, while negative slopes indicate it declines.

In the exercise, we have the equation \( f(x) = -6x \), where the slope \( m = -6 \). This shows that for every increase by 1 in \( x \), the value of \( y \) decreases by 6. This negative slope implies the line is heading downward as it moves from left to right. Understanding this change can help you predict how the line behaves, even before graphing it.

Finding the Y-Intercept

The y-intercept is where the line crosses the y-axis. In the linear equation format \( y = mx + b \), the y-intercept is represented by \( b \). It is the value of \( y \) when \( x \) is zero. This is an essential part of graphing because it gives you a starting point to draw your line. In our example, \( f(x) = -6x \), there is no constant term. This means the y-intercept \( b = 0 \).So, the line intersects the y-axis at the origin, at point \((0,0)\). This is where you'll start plotting when drawing the graph of the equation. Knowing \( b \) allows you to confidently place the first dot on your graph and begin to sketch the line.

Graphing Linear Equations

Graphing linear equations is all about connecting points that satisfy the equation, creating a visual representation of how \( x \) and \( y \) relate. Here's a straightforward way to graph linear equations such as \( f(x) = -6x \):

• Identify the y-intercept from the equation. For \( f(x) = -6x \), start at point \((0,0)\) on the y-axis.
• Use the slope to find another point. Each time you move horizontally 1 unit to the right (the "run"), go down 6 units (the "rise" becomes negative due to the slope being -6).
• Plot this second point at \((1, -6)\).
• Draw a straight line through these plotted points. Ensure the line extends across the grid, maintaining the slope ratio.

By following these steps, you'll have a clear and accurate graph of the linear equation. The line represents every solution to the equation, giving you a visual guide to the relationship expressed mathematically by the linear equation.