Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Match each equation with the graph in choices A-H that it most closely resembles. A.(GRAPHS CANNOT COPY) B.(GRAPHS CANNOT COPY) C.(GRAPHS CANNOT COPY) D.(GRAPHS CANNOT COPY) E.(GRAPHS CANNOT COPY) F.(GRAPHS CANNOT COPY) G.(GRAPHS CANNOT COPY) H.(GRAPHS CANNOT COPY) $$y=-3$$

Step-by-Step Solution

Answer

The graph for \( y = -3 \) is a horizontal line crossing the y-axis at -3.

Step 1: Understand the Equation

The equation given is \( y = -3 \), which is a horizontal line on the Cartesian plane.

Step 2: Analyze the Graph Characteristics

A graph of the equation \( y = -3 \) is a straight horizontal line that crosses the y-axis at -3. It does not slope upwards or downwards, since it is perfectly horizontal. This means it runs parallel to the x-axis.

Step 3: Match the Equation to the Graph

Look for the graph choice that shows a horizontal line crossing the y-axis at -3. It should not cross any other points on the y-axis and must not be sloped. Compare this graph against others to ensure it is the correct representation of \( y = -3 \). Select the graph among choices A-H that corresponds to this description.

Key Concepts

Understanding Horizontal Lines

A horizontal line is a type of line that runs from left to right and remains completely flat across the plane. It is characterized by having no vertical change as you move along the line. This unique feature makes it parallel to the x-axis, distinguishing it from other lines which may slope upwards or downwards.

In the context of a linear equation, a horizontal line has an equation of the form \( y = c \), where \( c \) is a constant. This constant represents the point where the line intersects the y-axis. For example, if \( c = -3 \), the horizontal line will cross the y-axis at \( -3 \).

Some key points about horizontal lines include:

• They have a slope of 0, indicating no vertical movement.
• They are perpendicular to vertical lines.
• In any horizontal line equation, the y-coordinate is constant while the x-coordinate can vary.

Understanding these properties will help in correctly identifying and graphing horizontal lines.

Exploring the Cartesian Plane

The Cartesian plane is a two-dimensional flat surface where all linear equations are plotted. Named after the mathematician René Descartes, it provides a framework to graphically represent relationships between two variables.

The Cartesian plane consists of two axes:

• x-axis: A horizontal line that extends left and right.
• y-axis: A vertical line that extends up and down.

These axes intersect at a point called the origin, denoted as \( (0, 0) \), and they divide the plane into four quadrants.

The Cartesian plane is essential for visualizing equations, as it allows us to see how one variable influences another. By plotting points or drawing lines based on given equations, we gain deeper insights into the relationships described by those equations. Simply put, this plane is our canvas for exploring all kinds of linear and non-linear equations.

The Role of the Y-Axis

The y-axis is a fundamental part of the Cartesian plane. It represents the vertical direction on the graph. Every point on the y-axis has an x-coordinate of zero, meaning these points exclusively measure distance above or below the origin.

In the context of graphing linear equations, the y-axis is crucial because it helps determine where lines intersect and their vertical positioning. For a horizontal line, intersections with the y-axis are particularly telling. The equation \( y = -3 \) demonstrates this well, showing that the line crosses exactly at \( -3 \) on the y-axis.

Key characteristics of the y-axis include:

• It separates the left and right sides of the Cartesian plane.
• Positive values rise above the origin, while negative values fall below.
• It helps measure vertical change between points on a graph.

Understanding how the y-axis functions is essential when plotting any line or point within the Cartesian grid. It provides the reference needed to accurately fix a graph on the plane.