Problem 84 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graph of each equation. $$ r(x)=4 $$

Step-by-Step Solution

Answer

A horizontal line at \( y = 4 \).

Step 1: Identify the Type of Equation

The given equation is \( r(x) = 4 \). This indicates a horizontal line on the Cartesian plane, as it is in the form \( y = k \) where \( k \) is a constant.

Step 2: Determine the Characteristics of the Graph

Since \( r(x) = 4 \) represents a horizontal line, it intersects the y-axis at \( y = 4 \). The line is parallel to the x-axis and every point on the line has a y-coordinate of 4.

Step 3: Plot the Intercept

On the Cartesian plane, locate the point (0, 4) on the y-axis, as this is the point where the line crosses the y-axis.

Step 4: Draw the Horizontal Line

Draw a straight horizontal line through the point (0, 4) that extends infinitely in both directions, which represents the graph of the equation.

Key Concepts

Horizontal Line

When we talk about horizontal lines in mathematics, we refer to lines that extend left and right, parallel to the x-axis, across the Cartesian plane.
A horizontal line does not tilt upwards or downwards, maintaining a constant elevation. An equation like \( r(x) = 4 \) illustrates this perfectly, where \( r(x) = 4 \) can be rewritten as \( y = 4 \).
This means every point on the line has the same y-value of 4 regardless of its x-coordinate.

• The main feature of a horizontal line is that it has a slope of 0.
• Slope's calculated as rise over run, and since there's no rise (change in y), it's always 0 for horizontal lines.
• These lines are key in graphing because they quickly tell you about the y-coordinate for all points on the line.

Understanding horizontal lines helps when sketching graphs and interpreting equations, especially in understanding what remains unchanged—namely, the y-coordinate.

Cartesian Plane

The Cartesian plane is a two-dimensional graphing framework that allows us to plot mathematical functions and equations visually.
It's made up of two perpendicular lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically.

• The origin is the point where the x-axis and y-axis intersect, denoted as (0,0).
• Each point on the plane is represented by a pair of coordinates (x, y).
• Understanding the Cartesian plane is crucial for graphing because it establishes a universal system for pinpointing locations.

  It acts as a visual tool that aids in understanding how equations represent shapes and lines, like identifying the zero slope of horizontal lines and the locations of different y-intercepts.

Y-intercept

The y-intercept of a graph is where a line or curve crosses the y-axis on the Cartesian plane.
For our horizontal line equation \( r(x) = 4 \), the y-intercept is particularly straightforward.
This line crosses the y-axis at the point (0, 4).

• Y-intercepts are helpful because they are starting points when sketching a graph.
• They indicate the value of \( y \) when \( x \) is zero.
• For horizontal lines, every point on the line shares the same y-coordinate, which is the intercept.

By pinpointing the y-intercept, you can more easily understand the nature and position of the line. This concept is foundational for graphing as it often defines where the line is anchored on the graph.