Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=1-2 x$$

Step-by-Step Solution

Answer

The linear function is decreasing with a y-intercept at (0,1) and passes through (1,-1).

Step 1: Identify the Type of Function

The given function is linear and of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( f(x) = 1 - 2x \) can be rewritten as \( f(x) = -2x + 1 \). Thus, the slope \( m = -2 \) and y-intercept \( b = 1 \).

Step 2: Plot the Y-intercept

Locate the point on the y-axis where the graph intersects it. This point is \((0, 1)\) because \( b = 1 \).

Step 3: Use the Slope to Find Another Point

The slope \( m = -2 \) means for every 1 unit increase in \( x \), \( f(x) \) decreases by 2 units. From the point \((0, 1)\), move 1 unit to the right (to \(x = 1\)) and 2 units down (to \(y = -1\)). This gives the point \((1, -1)\).

Step 4: Draw the Line

Use a ruler to draw a straight line through the points \((0, 1)\) and \((1, -1)\). Extend the line in both directions and add arrows at the ends to show that it continues indefinitely.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial when working with linear functions. This form is expressed as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. In the equation \( f(x) = -2x + 1 \), the slope \( m = -2 \) and the y-intercept \( b = 1 \), making the function easy to interpret and graph. The slope indicates how much the function increases or decreases as you move along the x-axis, while the y-intercept shows where the line crosses the y-axis. Equations written in this form allow for straightforward graphing and analysis, as the constants \( m \) and \( b \) directly provide key information about the line's behavior. When you have the equation in slope-intercept form, you can immediately identify these key characteristics, which simplifies both plotting the graph and understanding the relationship it represents. This form is not only convenient but also the most common way to represent linear equations in elementary mathematics courses.

Graphing Linear Equations

Graphing linear equations involves plotting points that satisfy the line's equation and drawing a line through them. For the function \( f(x) = -2x + 1 \), start by identifying the y-intercept, then use the slope to find another point.

• Identify the y-intercept: The equation gives a y-intercept of \((0, 1)\).
• Use the slope: The slope of \(-2\) means a change of 1 in \( x \) results in a change of \(-2\) in \( y \). From \((0, 1)\), move right by 1 unit to \( x=1 \) and down 2 units to \( y=-1 \), identifying the point \((1, -1)\).
• Draw the line: With a ruler, connect \((0, 1)\) and \((1, -1)\). Extend the line in both directions, adding arrows to indicate it continues infinitely.

This straightforward method for graphing makes it easy to visualize linear relationships, providing a tangible understanding of the equation's impact.

Y-Intercept

The y-intercept is a vital concept in understanding and graphing linear functions. It is the point where the graph of the function crosses the y-axis, and it occurs when \( x = 0 \). In the function \( f(x) = -2x + 1 \), the y-intercept is \( b = 1 \), so the coordinate is \((0, 1)\). Knowing the y-intercept allows you to start graphing the line without any calculations, as it is directly provided by the equation. From this starting point, you can apply the slope to plot additional points. The y-intercept provides a clear initial fixed reference on the graph, facilitating the drawing of the entire linear function.This intersection is not just a number; it holds a valuable role in illustrating where the function starts on the y-axis, and it can be easily spotted on a graph, making it a practical tool in analysis and computation.