Problem 25 - Brief 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.] $$ f(x)=-x+1 $$

Step-by-Step Solution

Answer

A straight line with slope -1, crossing y-axis at (0, 1).

Step 1: Identify the type of function

The function given is in the form of a linear equation, which is typically written as \( y = mx + b \). In this case, the function is \( f(x) = -x + 1 \), where \( m = -1 \) and \( b = 1 \). This tells us the slope is \(-1\) and the y-intercept is \(1\).

Step 2: Plotting the y-intercept

The y-intercept of the function is the point where the graph crosses the y-axis. This occurs at \( x = 0 \). Substitute \( x = 0 \) into the function: \( f(0) = -0 + 1 = 1 \). So plot the point \( (0, 1) \).

Step 3: Use the slope to find another point

The slope \( m = -1 \) tells us that for every unit movement to the right along the x-axis, the y-value decreases by 1 unit. From the y-intercept \( (0, 1) \), move 1 unit to the right to \( x = 1 \) and 1 unit down to \( y = 0 \). This gives us the point \( (1, 0) \). Plot this point as well.

Step 4: Draw the line

Using the points determined, \( (0, 1) \) and \( (1, 0) \), draw a straight line through these points. This line represents the graph of the function \( f(x) = -x + 1 \).

Step 5: Verify with another point

To ensure accuracy, test another point. Choose \( x = 2 \): \( f(2) = -2 + 1 = -1 \). Plot \( (2, -1) \) on the graph. It should fall on the line, verifying its correctness.

Key Concepts

Slope-Intercept Form

When graphing linear functions, one of the most straightforward forms to use is the slope-intercept form. This form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.

• Slope \( (m) \): The slope indicates the steepness and direction of the line. For example, a positive slope moves upward, while a negative slope goes downward as you move from left to right.
• Y-Intercept \( (b) \): The y-intercept is the point where the line crosses the y-axis. This happens when \( x = 0 \).

In the function \( f(x) = -x + 1 \), the slope \( m \) is \(-1\) and the y-intercept \( b \) is \(1\). Knowing these values simplifies the process of graphing because they direct us on where to start (the y-intercept) and how to proceed (using the slope).

Y-Intercept

The y-intercept of a linear function is crucial for sketching its graph. It is the point where the line touches the y-axis. In any linear equation given in slope-intercept form, the y-intercept \( b \) is simply the constant term. For \( f(x) = -x + 1 \), the y-intercept is \( 1 \), meaning that the line crosses the y-axis at the point \( (0, 1) \).To find the y-intercept quickly:

• Set \( x = 0 \) in the equation. The y-value you get will be the intercept. For instance, substituting \( 0 \) in \( f(x) = -x + 1 \) gives \( f(0) = 1 \).

This step is always your first plotting point, marking the start of your line on a graph.

Plotting Points

Once you've identified the slope and y-intercept, plotting points for the line becomes straightforward. Plotting points refers to marking specific points on the graph that the line passes through. These points help guide the drawing of the accurate line.

• Start with the y-intercept: Locate it at \( (0, b) \). For \( f(x) = -x + 1 \), start at \( (0, 1) \).
• Use the slope: From the y-intercept, apply the slope \( m \) to find more points. Here, a slope of \(-1\) indicates moving one unit right and one unit down. Thus, from \( (0, 1) \), move to \( (1, 0) \).

Continuing this method, you can determine more points such as \( (2, -1) \). After plotting, draw a straight line through these points, ensuring they align perfectly. This graphical method verifies the behavior and correctness of the linear function.