Problem 24 - Applied Calculus

Question

Graph each function "by hand." [Note: Even if you have a graphing calculator, it is important to be able to sketch simple curves by finding a few important points.] $$ f(x)=2 x-3 $$

Step-by-Step Solution

Answer

The line has slope 2 and y-intercept -3, passing through (0, -3) and (1, -1).

Step 1: Understand the Function

The function provided is a linear equation of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this specific function, \(f(x) = 2x - 3\), the slope \(m\) is 2, and the y-intercept \(b\) is -3.

Step 2: Find the Y-intercept

The y-intercept is the point where the line crosses the y-axis. This occurs when \(x = 0\). Substitute \(x = 0\) into the equation: \(f(0) = 2(0) - 3 = -3\). So, the y-intercept is \((0, -3)\).

Step 3: Find Another Point

To determine the slope and draw the graph accurately, find another point on the line. Choose \(x = 1\): \(f(1) = 2(1) - 3 = -1\). Therefore, another point is \((1, -1)\).

Step 4: Plot the Points

On graph paper or a coordinate plane, plot the two points: \((0, -3)\) which is the y-intercept, and \((1, -1)\).

Step 5: Draw the Line

Connect the two plotted points with a straight line. Extend the line in both directions and add arrows to indicate that it continues indefinitely.

Key Concepts

Linear Equations

Linear equations are one of the basic building blocks in algebra. They represent a relationship between two variables, usually in the form of \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept.
These equations are called "linear" because when graphed, they form a straight line. Understanding linear equations is crucial because they appear frequently in various mathematical contexts and real-world scenarios.

• Formula: \(y = mx + b\)
• Slope (\(m\)): Indicates how steep the line is
• Y-intercept (\(b\)): The point where the line crosses the y-axis

A deep understanding of linear equations helps with not only graphing but also solving complex problems in algebra.

Slope and Y-Intercept

The slope and y-intercept are two key concepts that help define the characteristics of a line.
**Slope (\(m\))**: The slope indicates the steepness or incline of a line. It is calculated as the "rise" (change in \(y\)) over the "run" (change in \(x\)). A larger slope means a steeper line, while a smaller slope indicates a more gradual incline. In the equation \(f(x) = 2x - 3\), the slope \(m\) is 2, meaning the line rises by 2 units for every 1 unit it moves to the right.
**Y-intercept (\(b\))**: The y-intercept is the point where the line crosses the y-axis. For the equation \(f(x) = 2x - 3\), the y-intercept \(b\) is -3. This means the line crosses the y-axis at the point \((0, -3)\).
Together, the slope and y-intercept provide all the information needed to graph a line and understand its behavior.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph equations, like the one provided in the exercise. It consists of two axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which is designated as \((0, 0)\).
Each point on the coordinate plane is identified with an ordered pair \((x, y)\), where \(x\) denotes the position relative to the x-axis and \(y\) denotes the position relative to the y-axis. Using this method, we can locate and plot points, draw lines, and graph functions.

• The x-axis is horizontal
• The y-axis is vertical
• The origin is \((0, 0)\)

Understanding the layout of the coordinate plane is essential for graphing functions and visualizing algebraic solutions.

Plotting Points

Plotting points on the coordinate plane is a fundamental skill needed to graph linear functions. This involves identifying specific values on the x-axis and y-axis and marking their intersection on the plane. Plotting points accurately is important as it helps in defining the line precisely.
To begin plotting points, start by determining specific x-values and calculating the corresponding y-values using the given linear equation. For instance, with the function \(f(x) = 2x - 3\), when \(x = 0\), \(y = -3\), resulting in the point \((0, -3)\). When \(x = 1\), \(y = -1\), resulting in the point \((1, -1)\).

• Choose a value for \(x\)
• Use the equation to find \(y\)
• Mark the point \((x, y)\) on the plane

Once you've plotted these points, connect them with a straight line to represent the linear function graphically.