Problem 78 - 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. $$ f(x)=-5 x+4 $$

Step-by-Step Solution

Answer

Slope \( m = -5 \), y-intercept \( b = 4 \); graph line through (0, 4) and (1, -1).

Step 1: Identify the equation's form

The given linear equation is \( f(x) = -5x + 4 \). This is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.

Step 2: Extract the slope

From the equation \( f(x) = -5x + 4 \), we compare it with the slope-intercept form \( y = mx + b \). The coefficient of \( x \) is the slope \( m \). Thus, the slope \( m = -5 \).

Step 3: Determine the y-intercept

In the equation \( f(x) = -5x + 4 \), the constant term is \( 4 \). This represents the \( y \)-intercept \( b \) when the equation is in the form \( y = mx + b \). Therefore, the \( y \)-intercept \( b = 4 \).

Step 4: Graph the line using slope and y-intercept

Begin by plotting the \( y \)-intercept on the graph at the point \((0, 4)\). From this point, use the slope to determine another point. The slope \( m = -5 \) means rise over run is -5, or you drop 5 units for every 1 unit you move to the right. Plot another point at \((1, -1)\). Draw a line through these points to complete the graph of the equation.

Key Concepts

Slope-intercept form

In mathematics, the slope-intercept form is a way of writing the equation of a straight line. This form is essential for easily identifying and graphing linear equations. Specifically, it takes the form \( y = mx + b \), where:

• \( y \) represents the dependent variable (typically a function of \( x \))
• \( m \) stands for the slope of the line
• \( x \) is the independent variable
• \( b \) is the \( y \)-intercept, the point where the line crosses the \( y \)-axis

The main advantage of the slope-intercept form is that it makes it straightforward to read off both the slope and the \( y \)-intercept from the equation. The slope \( m \) gives you information about the steepness and direction of the line, while \( b \) indicates where the line meets the \( y \)-axis. This simple form makes graphing linear equations a more manageable process.

Graphing linear equations

Graphing linear equations involves plotting the line that represents the equation on a coordinate plane. To do this, you can start by identifying key features of the line, such as the slope and the \( y \)-intercept. Here's a step-by-step approach:

• First, ensure the equation is in slope-intercept form \( y = mx + b \).
• Locate the \( y \)-intercept, which is the point \((0, b)\), on the graph.
• Using the slope \( m \), determine the rise and run to find another point on the line. For instance, a slope of \(-5\) indicates moving down 5 units for every 1 unit you move to the right.
• Draw a straight line through these points to represent the linear equation.

This process helps visualize how changes in \( m \) and \( b \) will alter the graph's appearance, showing you how many different lines you can create with various slopes and intercept values.

Slope

The slope of a line is a crucial concept in understanding how it behaves. Represented by \( m \) in the equation \( y = mx + b \), the slope indicates the line's steepness and direction. Here's what you need to know:

• The slope represents the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
• A positive slope suggests the line rises as you move from left to right, while a negative slope suggests it falls.
• A slope of zero indicates a horizontal line, whereas an undefined slope implies a vertical line.

For example, in the equation \( f(x) = -5x + 4 \), the slope is \(-5\). This means for every unit you move to the right, the line drops down by 5 units, creating a steep, downward line. Understanding slope allows you to easily predict and graphically represent the orientation of a line.

y-intercept

The \( y \)-intercept is a fundamental part of a line's equation, crucial for understanding where the line starts on the \( y \)-axis. In the slope-intercept form \( y = mx + b \), \( b \) indicates the \( y \)-intercept.

• The \( y \)-intercept is the point where the line crosses the \( y \)-axis, typically at \((0, b)\).
• This base point is where \( x \) equals zero, providing a starting position for graphing the line.
• In practice, knowing the \( y \)-intercept simplifies drawing the graph as it gives a fixed initial point.

For instance, in the equation \( f(x) = -5x + 4 \), the \( y \)-intercept is \( 4 \), meaning the line crosses the \( y \)-axis at the point \((0,4)\). This provides a quick reference point needed to begin drawing the line accurately.