Problem 25 - 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

Graph the line by plotting the points (0, 1) and (1, 0), then draw the line through them.

Step 1: Identify the Function Type

The given function is in the form of a linear equation, \(f(x) = -x + 1\), which represents a straight line.

Step 2: Determine the Slope and Y-Intercept

Rewrite the function to identify the slope and y-intercept. The function \(f(x) = -x + 1\) is equivalent to \(y = -x + 1\). Here, the slope \(m\) is \(-1\) and the y-intercept \(c\) is \(1\).

Step 3: Plot the Y-Intercept

Start by plotting the y-intercept on the graph. Since \(c = 1\), place a point on the y-axis at \( (0, 1) \).

Step 4: Use the Slope to Find Another Point

The slope \(-1\) means that for every one unit increase in \(x\), \(y\) decreases by one unit. From the y-intercept \((0,1)\), moving one unit right in the positive \(x\)-direction to \((1, ? )\), you move one unit down, which lands you at \( (1, 0) \). Plot this point.

Step 5: Draw the Line

Connect the points \((0, 1)\) and \((1, 0)\) with a straight line. This line represents the graph of \(f(x) = -x + 1\). The line should extend in both directions indefinitely, following the pattern of the slope.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations, making it easy to graph them. It is written as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This format provides clear information about the inclination of the line and where it crosses the y-axis. In the function \(f(x) = -x + 1\), we can see that it follows the slope-intercept form by comparing it to the general structure \(y = mx + c\). This helps us quickly identify both the slope and the y-intercept without needing lengthy calculations.

Y-Intercept

The y-intercept is the unique point where the line crosses the y-axis. In the slope-intercept form \(y = mx + c\), the y-intercept is represented by \(c\). For the equation \(f(x) = -x + 1\), the y-intercept is \(1\). This tells us that the line will cross the y-axis at the point \((0, 1)\). When graphing, the first step is to plot this point on the y-axis. This forms the foundation for sketching the rest of the line, as everything else will be plotted relative to this point.

Slope

The slope indicates the steepness and direction of a line. It is represented by \(m\) in the slope-intercept form \(y = mx + c\). In \(f(x) = -x + 1\), the slope \(m\) is \(-1\). The negative sign means the line descends as it moves to the right. Specifically, for every unit increase in \(x\), \(y\) will decrease by the same amount, creating a diagonally downward line. Knowing the slope allows us to find multiple points on the graph by starting at the y-intercept and making consistent moves according to the slope value.

Linear Function Plotting

Plotting a linear function involves utilizing the information from the slope-intercept form to sketch the accurate graph. Begin by plotting the y-intercept point obtained from \(c\). For \(f(x) = -x + 1\), start at \((0, 1)\). Next, use the slope to find another point. With a slope of \(-1\), move one unit in the positive \(x\)-direction while moving one unit down, reaching \((1, 0)\). Mark this second point. Finally, draw a straight line through these points, extending it indefinitely in both the positive and negative directions. This results in a visual representation of the linear function, showcasing how it behaves across the graph.