Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Graph each equation by hand. $$y=5-10 x, y=|5-10 x|$$

Step-by-Step Solution

Answer

Draw a straight line for \( y = 5-10x \). For \( y = |5-10x| \), make a V-shape using the point (0.5, 0) as the vertex.

Step 1: Identify Forms of Given Equations

The first equation is a linear equation of the form \( y = mx + c \), where \( m = -10 \) and \( c = 5 \). The second equation involves an absolute value function, which modifies the linear equation into an absolute value equation \( y = |5 - 10x| \).

Step 2: Graph the Linear Equation

Graph the equation \( y = 5 - 10x \). This is a straight line with a y-intercept at (0,5) and a slope of -10, meaning it drops 10 units for every 1 unit it moves to the right.

Step 3: Plot the Line's Important Points

Find two points from the linear equation to draw the line accurately. Use the y-intercept (0, 5). Find another point by setting \( y = 0 \) and solve for \( x \): \( 5 - 10x = 0 \) gives \( x = 0.5 \). So, another point is (0.5, 0).

Step 4: Draw the Absolute Value Graph

The graph of \( y = |5 - 10x| \) is derived from the line \( y = 5 - 10x \) by reflecting the negative portion of the line over the x-axis. Plot the vertex of the V-shape, which occurs where \( 5 - 10x = 0 \), at \( (0.5, 0) \).

Step 5: Apply the V-Shape of Absolute Functions

Draw the V-shape by using the vertex at \( (0.5, 0) \). The portion of the line where \( y = 5 - 10x \) is positive remains unchanged, while the negative values are mirrored across the x-axis. Extend the lines from the vertex in both directions with the same slopes, \( -10 \) to the left and \( 10 \) to the right of \( x = 0.5 \).

Step 6: Verify the Graph

Check key points on both graphs. For \( x = 0 \), \( y = 5 \) for both equations. For \( x = 1 \), \( y = -5 \) for \( y = 5-10x \) and \( y = 5 \) for \( |5 - 10x| \). These points should match the graph.

Key Concepts

Linear Equation

A linear equation forms the backbone of graphing linear functions. It can be represented in the form of \( y = mx + c \), where \( m \) is the slope and \( c \) the y-intercept. Linear equations create straight lines when graphed on a coordinate plane.
The equation \( y = 5 - 10x \) is an example. It is a simple line that shows the relationship between \( x \) and \( y \).
Due to its predictable nature, it is widely used in elementary algebra.

• Each point \( (x, y) \) on the line satisfies the equation.
• The line continues infinitely in both directions unless restricted by a specific domain.
• The y-intercept is the point where the line crosses the y-axis.
• The slope decides the steepness and direction of the line.

Absolute Value

Absolute value equations add a twist to linear relationships by introducing the concept of magnitude, which makes every output positive. In essence, an absolute value, denoted by vertical bars as in \( |x| \), measures how far a number is from zero without considering its direction.
It has profound effects on graphing since negative parts "flip" upwards, creating a distinctive V-shape.

Here's how it changes equations:

• In the absolute value equation \( y = |5 - 10x| \), sections beneath the x-axis reflect upwards.
• This reflection creates two mirror images of the portion of the line above the x-axis.
• The vertex at \( (0.5, 0) \) is the point where the graph switches from negative to positive.
• The graph now appears as two intersecting lines stemming from the vertex.

Understanding this helps recognize graphs involving absolute functions immediately.

Slope

The slope is a crucial concept in graphing linear equations as it determines how a line spans across the coordinate plane. Defined as "rise over run," it compares vertical movement to horizontal movement. In the equation \( y = 5 - 10x \), the slope is \(-10\). This means the line falls steeply downward. For any 1 unit increase in \(x\), \(y\) decreases by 10 units.

• A positive slope slopes upwards as \(x\) increases.
• A negative slope, like in this case, slants downwards, creating a descending line.
• A greater numerical value in the slope means a steeper line.
• If the slope is zero, the line is horizontal.

Recognizing the impact of the slope makes graph prediction much easier. It serves as a visual cue to the direction and rate of change on a graph.

Y-intercept

The y-intercept serves as the point where a line crosses the y-axis, providing information about a graph's starting point vertically. For the linear equation \( y = 5 - 10x \), the y-intercept is \(5\). This is the value of \(y\) when \(x\) is zero.

• The y-intercept provides a starting point for graphing a line.
• It is marked by the coordinates \((0, c)\), where \(c\) is the intercept value.
• On a graph, it often indicates an initial condition or beginning state.
• The y-intercept can be quickly found by inspecting the equation's constant term.

Understanding this helps initiate sketching a line graph and builds a complete understanding of how the equation behaves.