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

Step-by-Step Solution

Answer

Graph the line using the points (0, -2) and (1, 1).

Step 1: Identify the Type of Function

The function given is a linear function, represented by the equation \( f(x) = 3x - 2 \). Linear functions graph as straight lines.

Step 2: Determine the Y-Intercept

The y-intercept of a function is the point where the graph crosses the y-axis (where \( x = 0 \)). Substitute \( x = 0 \) into the function: \( f(0) = 3 \times 0 - 2 = -2 \). So, the y-intercept is \( (0, -2) \).

Step 3: Determine the Slope

The slope of the line can be identified from the function in the form \( f(x) = mx + b \), where \( m \) is the slope. Here, \( m = 3 \), meaning the slope is 3. This indicates for every unit increase in \( x \), \( f(x) \) increases by 3 units.

Step 4: Plot the Y-Intercept

Begin graphing by plotting the y-intercept \( (0, -2) \) on the coordinate plane.

Step 5: Use the Slope to Find Another Point

From the y-intercept \( (0, -2) \), use the slope to find another point. With a slope of 3, move 1 unit right (to \( x = 1 \)) and 3 units up (since the slope 3 is \( \frac{3}{1} \)). This gives us the point \( (1, 1) \). Plot this point.

Step 6: Draw the Line

Using a ruler, draw a straight line through the points \( (0, -2) \) and \( (1, 1) \). Extend this line across the graph to represent the function.

Step 7: Check for Additional Points

Optionally, choose other values of \( x \) to ensure the accuracy of the line. For example, with \( x = 2 \), \( f(2) = 3 \times 2 - 2 = 4 \), so \( (2, 4) \) should be another point on the line. Confirm it fits your line.

Key Concepts

y-intercept

The y-intercept is a key concept when graphing linear functions. It is the point where the graph crosses the y-axis. This means it occurs when the value of x is zero. To find the y-intercept from the equation

• Substitute zero for x in the equation.
• Calculate the resulting value of the function.
• This result will be the coordinate for the y-intercept, in the form (0, y).

For instance, in the function \( f(x) = 3x - 2 \), replacing x with 0 gives us \( f(0) = -2 \). Therefore, the y-intercept is at the point (0, -2). Knowing the y-intercept helps to quickly locate where the line begins on the vertical axis of the coordinate plane.
Moreover, it provides a starting point for the graph when sketching the line.

slope

The slope of a linear function is a measure of its steepness and direction. Denoted by "m" in the slope-intercept form of the equation \( y = mx + b \), it indicates how much the y-value of the function changes for each unit increase in x. In other words, the slope tells us how much the line rises or falls as we move horizontally.

• A positive slope means the line ascends from left to right.
• A negative slope indicates the line descends.
• A slope of zero signifies a horizontal line.

In the example \( f(x) = 3x - 2 \), the slope is 3. This means for every 1 unit increase in x, the y-value increases by 3 units. Understanding the slope helps in drawing accurate graphs by determining points along the line based on the initial position of the y-intercept.

coordinate plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface used to graph functions. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at the origin (0,0).Every point on the plane can be described by an ordered pair (x, y), representing its horizontal and vertical positions, respectively. The plane is divided into four quadrants, with positive and negative values for both axes providing unique positions for every coordinate.When graphing a function like \( f(x) = 3x - 2 \), we use the coordinate plane to plot points such as the y-intercept (0, -2) and any other points derived using the slope. This allows us to see the relationship and pattern of the function visually.
The coordinate plane is essential because it translates algebraic equations into clear visual representations.

sketching functions

Sketching functions is a useful skill, especially for linear functions. This process requires understanding the key features of a function's equation and translating them into a graph on the coordinate plane.To sketch a linear function like \( f(x) = 3x - 2 \), you should start by:

• Identifying the y-intercept and plot it on the y-axis.
• Using the slope to find additional points.
• Drawing a straight line through the points using a ruler or straightedge.

It's helpful to check the graph by substituting other x-values into the equation to find corresponding y-values, ensuring the line represents the function accurately.
Sketching functions not only aids in visualizing mathematical concepts but also helps in checking work done with equations. It reinforces the understanding of fundamental concepts in algebra.