Problem 42 - Thinking Mathematically

Question

Graph each horizontal or vertical line. \(y=2\)

Step-by-Step Solution

Answer

The graph of the given equation \(y = 2\) is a horizontal line passing through the point (0,2).

Step 1: Identify the nature of line

First, identify the nature of the line. The general equation for a horizontal line is \(y = k\) where 'k' is a constant. Comparing this with the given equation \(y = 2\), it is clear that this is a horizontal line.

Step 2: Identify the y-intercept

Next, identify the y-intercept. The y-intercept is the point where the line intersects the Y-axis. In the given equation, '2' is the constant, which means the line intersects the Y-axis at the point (0,2).

Step 3: Plot the line

To plot the line, draw a straight line parallel to the X-axis and passing through the point on the Y-axis identified in the previous step (0,2). All points on this line will have the y-coordinate '2', regardless of the x-coordinate. This includes, for example, points like (-2,2), (0,2) and (2,2).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric figures like points, lines, and shapes using algebraic equations in a coordinate system. The most common system is the two-dimensional Cartesian coordinate system, where the horizontal axis is labeled as the 'x-axis' and the vertical as the 'y-axis'.

Each point in this system is defined by an ordered pair of numbers \( (x, y) \), where 'x' represents the horizontal position and 'y' stands for the vertical position. To visualize the concept of a horizontal line in coordinate geometry, such as \( y = 2 \), every point on this line has the same 'y' coordinate, in our case, '2'. No matter how far you travel along this line to the left or right (along the x-axis), the 'y' value remains constant, visually creating a straight line parallel to the x-axis.

Y-Intercept

The y-intercept of a line is the point where the line crosses or intersects the y-axis of a coordinate plane. If you look at the equation of a line in the slope-intercept form, \( y = mx + b \) where 'm' is the slope and 'b' is the y-intercept, you'll notice \( b \) represents the value of 'y' when \( x \) is zero. This is why, for a horizontal line like \( y = 2 \) mentioned in our exercise, the y-intercept is at the point (0, 2).

Understanding the y-intercept is crucial as it gives us an anchor point for graphing the entire line. It's where you begin plotting on the graph because you know exactly where the line will touch the y-axis, and from there, you can draw the rest of the line using the slope, or in the case of a horizontal line, by moving straight left and right.

Linear Equations

Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. These equations graph into straight lines, hence the name 'linear'. The simplest linear equation is that of a horizontal or vertical line. For a horizontal line, the equation simplifies to \( y = k \), with 'k' being a constant value.

Our exercise gives us the equation \( y = 2 \), which is a prime example of a linear equation representing a horizontal line. All the points on this line have a constant 'y' value of '2' and an 'x' value that can be any real number. In coordinate geometry, understanding linear equations is fundamental as it enables you to predict how changes in the equation affect the position and slope of the line on a graph.