Problem 50 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$y=-3$$

Step-by-Step Solution

Answer

A graph of the equation \(y = -3\) is a horizontal line passing through the point (0,-3) on the y-axis.

Step 1: Understand the Equation

The equation given is \(y = -3\). This is a simple linear equation. The y-value is constant and equal to -3 for all x-values. That means, no matter what x is, y is always -3. It's a horizontal line in the Cartesian coordinate system.

Step 2: Plot the Line

For the line \(y = -3\), it passes through the point (0,-3) on the y-axis. Therefore, draw a straight, horizontal line through the point (0,-3). Ensure the line extends in both directions (left and right), as there is no limit on the value of x.

Key Concepts

Cartesian Coordinate System

The Cartesian coordinate system is a foundational tool in algebra, providing a framework to graphically represent equations on a two-dimensional plane. This system consists of two perpendicular lines, typically referred to as axes. The horizontal axis is known as the x-axis, and the vertical is the y-axis. Every point on this plane is represented by an ordered pair \( (x, y) \) indicating its horizontal and vertical positions respectively. For example, the origin \( (0, 0) \) is the point where both axes intersect. When graphing an equation such as \( y = -3 \), it's crucial to understand how the coordinates relate to the line. Here, the consistent presence of '-3' as the y-value means that the line will traverse horizontally across the plane, touching every possible x-value at a vertical position of -3.

Horizontal Lines in Algebra

In algebra, horizontal lines have a crucial characteristic: their y-value remains constant no matter what the x-value is. To visualize this, imagine walking endlessly to your left or right while staying on the same step of a staircase – you are essentially drawing a horizontal line at that step’s level. In a Cartesian coordinate system, a horizontal line is described by an equation of the form \( y = k \), where \( k \) is a constant. This means that every point along that line has the same y-coordinate. So, for the equation \( y = -3 \) we’re considering, regardless of whether \( x \) is 1, 100, or -50, \( y \) will always be -3. Graphing a horizontal line is simple: locate the \( y \) value on the y-axis— that’s your starting point and then draw a straight line to the left and right, keeping the line completely flat and parallel to the x-axis.

Constant Function

A constant function in algebra is a special type of linear equation where the output does not vary despite changes in the input. In simpler terms, for the function \( y = c \), no matter what value of \( x \) is chosen, the value of \( y \) will always be the constant \( c \). It’s like a flat road – whether you travel a mile or a hundred miles, the elevation stays the same. Our example \( y = -3 \) is a constant function because the y-value is fixed at -3. The significance of a constant function in graphing is that it’s depicted as a horizontal line in the Cartesian coordinate system. This visualization aids students in understanding how algebraic concepts translate into geometric representations, providing a clearer depiction of what a constant function 'looks like' mathematically.