Problem 77 - Calculus Volume 1

Question

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

Slope: -6, Y-intercept: 0. Graph through points (0, 0) and (1, -6).

Step 1: Identify the Linear Equation

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

Step 2: Determine the Slope

The equation \(f(x) = -6x\) can be represented as \(y = -6x + 0\). Thus, the slope \(m\) is \(-6\).

Step 3: Determine the Y-Intercept

In the equation \(y = -6x + 0\), the y-intercept \(b\) is \(0\). This means the line crosses the y-axis at the origin \((0, 0)\).

Step 4: Plot the Y-Intercept

To graph the line, start by plotting the y-intercept, which is the point \((0, 0)\).

Step 5: Use the Slope to Plot Another Point

The slope \(m = -6\) indicates a rise of \(-6\) units for each run of \(+1\) unit. Starting from \((0, 0)\), move 1 unit to the right (to \(x = 1\)) and 6 units down (to \(y = -6\)) to mark the point \((1, -6)\).

Step 6: Draw the Line

Connect the plotted points \((0, 0)\) and \((1, -6)\) with a straight line. Extend the line in both directions to complete the graph of the linear equation.

Key Concepts

Understanding the Slope

The slope, often denoted by the letter \(m\), is a crucial part of any linear equation. In simple terms, it measures the steepness or incline of a line. The slope tells us how the line rises or falls as you move along the x-axis.

• If the slope is positive, the line ascends (goes up) as it moves from left to right.
• If the slope is negative, the line descends (goes down) as it moves from left to right.
• A slope of zero indicates a flat, horizontal line.

In our example, the linear equation is \(f(x)=-6x\) or \(y=-6x\). The slope here is \(-6\). This means that for every 1 unit the line moves to the right, it drops 6 units down. Hence, the slope is negative, indicating a decreasing line. Understanding the slope helps us predict how the line will behave just by looking at the equation.

The Role of the Y-Intercept

The y-intercept, symbolized by \(b\), in a linear equation of the form \(y = mx + b\), is the point where the line crosses the y-axis. This point is significant because it gives us a starting point for graphing the line.

• The y-intercept is where \(x = 0\), hence it is represented by the point \((0, b)\).
• In the equation \(f(x) = -6x\), which can be rewritten as \(y = -6x + 0\), the y-intercept \(b\) is clearly \(0\).
• This tells us that the line crosses the y-axis exactly at the origin point \((0, 0)\).

Knowing the y-intercept allows us to graph the line efficiently by giving a known point from which to start.

Graphing Linear Functions Made Easy

Graphing linear functions involves plotting points on a graph and then connecting these points to form a straight line. Let's take a closer look at how to do this with our function \(f(x) = -6x\):

• Begin by plotting the y-intercept. As identified, our starting point is \((0, 0)\), because the y-intercept is 0.
• Use the slope to determine another point on the line. The slope \(-6\) tells us to move 1 unit to the right and 6 units down. This takes us to the point \((1, -6)\).
• Once these points are plotted, draw a straight line through them. Extend the line across the graph to clearly see the linear relationship.

With practice, graphing these functions becomes straightforward, allowing you to visualize how changes in slope and y-intercept affect the position and angle of the line.