Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Graph each line by hand. Give the \(x\) - and y-intercepts. $$x-y=4$$

Step-by-Step Solution

Answer

The x-intercept is (4,0) and the y-intercept is (0,-4). Graph the line through these points.

Step 1: Rewrite the Equation in Slope-Intercept Form

Start by rewriting the given equation in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The original equation is \(x - y = 4\). Solve for \(y\) by subtracting \(x\) from both sides and then multiplying by -1: \(y = x - 4\).

Step 2: Identify the Y-Intercept

The y-intercept occurs where the line crosses the y-axis, which means \(x = 0\). Substitute \(x = 0\) into \(y = x - 4\) to find \(y\). \(y = 0 - 4 = -4\). Thus, the y-intercept is \((0, -4)\).

Step 3: Calculate the X-Intercept

The x-intercept occurs where the line crosses the x-axis, meaning \(y = 0\). Substitute \(y = 0\) into \(x - y = 4\) to find \(x\). \(x - 0 = 4\), so \(x = 4\). Thus, the x-intercept is \((4, 0)\).

Step 4: Graph the Line Using Intercepts

Plot the intercepts \((0, -4)\) and \((4, 0)\) on a Cartesian coordinate system. Draw a straight line through these two points. This line represents the equation \(x - y = 4\).

Step 5: Verify the Graph

Check the graph by ensuring it passes through both intercept points on the plot and its slope matches \(m = 1\) compared with \(y = x - 4\). This confirms the vertical rise to horizontal run is consistent between points.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a method to express the equation of a line in an easy-to-interpret format. This form is written as \( y = mx + b \), where:

• \( m \) represents the slope, indicating the steepness and direction of the line.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

When you rewrite an equation in slope-intercept form, you can quickly determine these critical characteristics of the line. The slope \( m \) shows us how much \( y \) changes with a change in \( x \). For each unit the line moves horizontally, it moves \( m \) units vertically. In our example, the equation \( x - y = 4 \) can be rearranged to \( y = x - 4 \). Here, the slope \( m \) is 1, telling us that the line rises one unit on the y-axis for each one unit it moves along the x-axis. This form allows for easy graphing and immediate visual insights into the line's characteristics.

X-Intercept

The x-intercept is a point where the line crosses the x-axis. At this exact point, the value of \( y \) is always zero. To find the x-intercept of an equation, set \( y = 0 \) and solve for \( x \).
In the original equation \( x - y = 4 \), to find the x-intercept, substitute the value of \( y \) as 0. This simplifies the equation to \( x - 0 = 4 \), resulting in \( x = 4 \). Thus, the x-intercept of our line is \( (4, 0) \).
It's a crucial point because it provides a tangible starting or ending point on the x-axis when graphing the line. This intercept interacts with the y-intercept to guide the placement of your line on a graph. Understanding intercepts helps in accurately sketching the line by providing key points necessary for its directional plotting.

Y-Intercept

The y-intercept is where the line crosses the y-axis in a coordinate plane. At this point, \( x \) equals zero. To locate the y-intercept, substitute \( x = 0 \) into your equation and solve for \( y \).
For the given equation \( y = x - 4 \), placing \( x = 0 \) provides \( y = 0 - 4 = -4 \). Hence, the y-intercept is \( (0, -4) \). This point is essential for understanding where your graph begins in relationship to the y-axis.
Alongside the slope, the y-intercept, \( b \), is a cornerstone in interpreting the line's equation. It shows the start point of the line's path on the vertical axis. Intercepts not only assist in graphing but also inform about the behavior of lines: a change here affects the line's elevation relative to the y-axis without impacting its direction.