Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Graph each equation by hand. $$y=4-8 x, y=|4-8 x|$$

Step-by-Step Solution

Answer

Graph \(y = 4 - 8x\) as a line, and \(y = |4 - 8x|\) as a 'V' centered at \((0.5, 0)\).

Step 1: Understanding the Linear Equation

The first equation is a linear equation of the form \( y = mx + b \), where \( m = -8 \) and \( b = 4 \). This implies that the slope of the line is \(-8\) and the y-intercept is \(4\). We'll graph this equation first as it is a straight line.

Step 2: Plot the Y-Intercept

Start by plotting the y-intercept on the graph. Since \( b = 4 \), mark the point \((0,4)\) on the y-axis.

Step 3: Use the Slope to Find Another Point

The slope \(-8\) means for every 1 unit you move to the right (positive x-direction), you move 8 units down (negative y-direction) from the y-intercept. From \((0,4)\), go 1 unit right and 8 units down to plot the next point at \((1, -4)\).

Step 4: Draw the Line

Using a straight edge, draw a line through the points \((0,4)\) and \((1,-4)\). This line represents the equation \( y = 4 - 8x \).

Step 5: Understand the Absolute Value Equation

The second equation \( y = |4 - 8x| \) involves an absolute value, transforming the graph of the linear equation. Absolute value turns any negative y-values into positive y-values, creating a 'V' shape.

Step 6: Graphing the Absolute Value Equation

Identify where the original line crosses the x-axis to determine the vertex of the 'V'. Set \( y = 0 \) in \( y = 4 - 8x \) to find \( x = 0.5 \). Plot a point at \((0.5, 0)\) and the original y-intercept \((0, 4)\). Reflect the part of the line below the x-axis over the x-axis to complete the 'V' shape.

Step 7: Finalize and Review the Graph

Ensure both graphs are clearly labeled: the line from \( y = 4 - 8x \) with a slope and the 'V' shaped graph from \( y = |4 - 8x| \). Note the significant points, including where lines intersect with axes and the vertex of the 'V'.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics. They describe a straight line on the graph. The structure of a linear equation is generally denoted as \( y = mx + b \). In this form, \( m \) is the slope, and \( b \) is the y-intercept. The slope represents the steepness of the line and the y-intercept is where the line crosses the y-axis.

For example, in the equation \( y = 4 - 8x \), the slope \( m \) is \(-8\), indicating a downward slant, while the y-intercept \( b \) is \(4\), locating the starting point of our graph.

• Start by plotting the intercept: on the y-axis at \((0,4)\).
• Next, apply the slope: move 1 unit on the x-axis and 8 units down on the y-axis.

These steps allow you to plot another point like \((1, -4)\), helping to form the line graph representing the equation.

Absolute Value Functions

Absolute value functions modify linear equations to display their behavior differently, introducing a distinct 'V' shape into the graph. The equation \( y = |4 - 8x| \) uses absolute value to alter standard lines.

An absolute value function takes any negative output from a linear equation and converts it to a positive, leaving positive values unchanged. This transformation introduces a vertex, a crucial point where the graph reflects.

For \( y = |4 - 8x|\):

• Determine the vertex by setting \( y = 0 \) in \( 4 - 8x \), which gives \( x = 0.5 \), leading to a point at \((0.5,0)\).
• Reflect the lower part if the line under the x-axis (negative y-values), creating symmetry.

This reflection helps in constructing the 'V' shape around the vertex providing a clear view of the absolute value function's behavior.

Slope and Intercept

The slope and intercept are integral components of linear equations, defining the line's direction and starting point on a graph. The slope \( m \) in \( y = mx + b \) indicates rise over run, which could be positive (upwards) or negative (downwards) moving.

In the equation \( y = 4 - 8x \), the slope of \(-8\) reflects a steep decline, while the intercept \( b = 4 \) shows the line crossing the y-axis at 4.

Key aspects to understand:

• The higher the absolute value of the slope, the steeper the line.
• The intercept defines the elevation at the y-axis start point, serving as a basis for plotting further points using the slope.

By leveraging both features, one can illustrate the entire trajectory of the equation graphically.

Graph Transformations

Graph transformations involve altering existing graphs of equations to reflect shifts or changes, often through operations like absolute value, translations, or scaling. Transformations are especially evident with absolute value functions, like \( y = |4 - 8x| \), affecting graph paths.

Simple transformations include:

• Vertical or horizontal shifts, translating whole graphs along the axes.
• Reflections, turning negative sections positive through operations like absolute value.
• Stretches or compressions, changing slope steepness, affecting the wide or narrow shape.

For instance, with \( y = |4 - 8x| \), a reflection across the x-axis flips negative y-axis values, creating the 'V' shape. Such transformations visually redefine a graph, aiding in better interpretation of mathematical functions.