Problem 78 - Calculus Volume 1

Question

For each linear equation, a. give the slope \(m\) and \(y\) -intercept \(b\), if any, and b. graph the line. \(\quad f(x)=-5 x+4\)

Step-by-Step Solution

Answer

Slope \( m \) is \(-5\) and y-intercept \( b \) is \( 4 \).

Step 1: Identify the Equation Form

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

Step 2: Determine the Slope

In the equation \( f(x) = -5x + 4 \), the coefficient of \( x \) is the slope. Therefore, the slope \( m \) is \(-5\).

Step 3: Determine the y-intercept

The constant term in the equation \( f(x) = -5x + 4 \) represents the y-intercept. Thus, the y-intercept \( b \) is \( 4 \).

Step 4: Graph the Line

To graph the equation, start by plotting the y-intercept \( (0, 4) \) on the coordinate plane. From this point, use the slope \( -5 \) to find another point. Since the slope \( -5 \) is equivalent to \( \frac{-5}{1} \), move down 5 units and right 1 unit from the point \( (0, 4) \) to find the next point, \( (1, -1) \). Draw a straight line through these points to represent the equation.

Key Concepts

Slope and y-intercept

When dealing with linear equations, it's crucial to understand the role of slope and y-intercept. The equation of a line in the form \( y = mx + b \) clearly distinguishes these components. Here, \( m \) represents the **slope**, and \( b \) indicates the **y-intercept**. The slope \( m \) determines the steepness and direction of the line. For example, a positive slope means the line ascends as you move from left to right, while a negative slope, like \(-5\) in our equation \( f(x) = -5x + 4 \), shows the line descends.

The y-intercept \( b \) is the point where the line crosses the y-axis. This is where \( x = 0 \). In our case, the y-intercept is \( 4 \).

Understanding both slope and y-intercept can help when quickly identifying and sketching graphs of linear equations.

Graphing Linear Equations

Graphing a linear equation like \( f(x) = -5x + 4 \) involves plotting points and drawing a line through them. Here's how you can efficiently graph such equations:

• Start with the y-intercept \( b \). For our equation, this point is \( (0, 4) \).
• Next, use the slope \( m \). The slope of \(-5\) tells you to move down 5 units for every 1 unit you move to the right. This helps you find a second point on the line.
• Continue locating more points if necessary or desired, using the slope.

Once you have at least two points plotted, draw a straight line through them. Extend the line across the grid to fully represent the equation.

It's as straightforward as that! Just remember, aligning the points well ensures an accurate graph.

Coordinate Plane

The coordinate plane is a vital tool for visualizing linear equations. It consists of two perpendicular lines called axes. The horizontal line is the **x-axis**, and the vertical line is the **y-axis**. Together, they divide the plane into four quadrants.

Whenever you graph a linear equation, such as \( f(x) = -5x + 4 \), you plot points using ordered pairs \((x, y)\). For example, the point \((0, 4)\) represents the y-intercept on this plane. Each value along the axes can be positive or negative, creating versatility in graphing different equations.

Understanding the coordinate plane helps you visually represent mathematical concepts accurately, enhancing comprehension and problem-solving skills in algebra. Whether plotting a simple line or tackling more complex functions, this tool is essential in any student's journey to mastering graphs.