Problem 23 - Brief Applied Calculus

Question

23-30. 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.] $$ f(x)=3 x-2 $$

Step-by-Step Solution

Answer

Plot the y-intercept at (0, -2), use the slope to find another point at (1, 1), and draw the line through these points.

Step 1: Identify the Function Type

The given function is a linear function of the form \( f(x) = mx + b \). For this particular function, \( m = 3 \) and \( b = -2 \). This tells us the slope (\( m \)) and the y-intercept (\( b \)).

Step 2: Plot the Y-intercept

The y-intercept is the point where the graph intersects the y-axis. For the function \( f(x) = 3x - 2 \), the y-intercept is \( (0, -2) \). Plot this point on the graph.

Step 3: Use the Slope to Find Another Point

The slope \( m = 3 \) means that for every unit increase in \( x \), \( f(x) \) increases by 3 units. From the y-intercept \( (0, -2) \), move 1 unit to the right (to \( x = 1 \)) and move up 3 units (to \( y = 1 \)). This gives another point on the line: \( (1, 1) \). Plot this point.

Step 4: Draw the Line

Using a ruler, draw a straight line through the two points \( (0, -2) \) and \( (1, 1) \). This line represents the graph of the function \( f(x) = 3x - 2 \).

Step 5: Verify with Additional Points

To ensure accuracy, calculate \( f(x) \) for another value of \( x \). For example, for \( x = 2 \), \( f(x) = 3(2) - 2 = 4 \). Plot the point \( (2, 4) \). It should lie on the line drawn.

Key Concepts

Linear Equations

Linear equations form the foundation of algebra and appear as straight lines when graphed. A linear equation is any equation that can be written in the form \( ax + by = c \), where \( x \) and \( y \) are variables, and \( a, b, \) and \( c \) are constants. A special feature of linear equations is that they produce straight-line graphs, which is why they are called "linear."

In our exercise, we dealt with the function \( f(x) = 3x - 2 \), which is already in the so-called slope-intercept form. Recognizing linear equations helps us predict the straight-line behavior of the graph. With linear equations, such as the one in the exercise, you need only two points to sketch the entire line. More complex functions, like quadratic or exponential ones, will not have this characteristic simplicity. Understanding this simplicity is crucial for successfully graphing linear functions "by hand."

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a line. It follows the format \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept. This form helps you quickly identify key characteristics of a linear equation.

• Slope (\( m \)): Indicates the steepness of the line. In the exercise, \( m = 3 \), meaning the line rises 3 units for each unit it moves to the right along the x-axis.
• Y-intercept (\( b \)): The point where the line crosses the y-axis. Here, \( b = -2 \), so the graph crosses at the point \((0, -2)\).

To graph a linear function using the slope-intercept form, you start by plotting the y-intercept and then use the slope to determine another point on the line. This method is straightforward and very helpful for students learning to hand-draw graphs. It offers a quick way to visualize what a linear function looks like.

Hand-Drawn Graphing

Hand-drawn graphing is a fundamental skill in mathematics. While technology, like graphing calculators, provides quick solutions, understanding how to graph by hand deepens comprehension. It involves visualizing mathematical relationships, which is why the exercise asks students to draw the graph manually.

In the case of \( f(x) = 3x - 2 \), we begin with the y-intercept point \((0, -2)\). The next step involves the slope, which tells you how to move from the y-intercept to another point. For a slope of 3, you move 1 unit on the x-axis and 3 units up on the y-axis to reach \((1, 1)\).

With these two points marked, you can draw a straight line through them, which is the graph of the function. Verifying with another point, like \((2, 4)\), ensures accuracy. This method highlights the systematic nature of mathematics and transforms abstract equations into tangible visuals on the graph. Mastering hand-drawn graphing allows students to build confidence in their mathematical understanding without relying on technological tools.