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Magazine Chapter 3: Problem 52
Graph each equation. $$ y=-3 $$
Short Answer
Expert verified
The graph is a horizontal line crossing the y-axis at \(-3\).
Step by step solution
01
Understanding the Equation
The equation given is \( y = -3 \). This represents a horizontal line in a coordinate plane.
02
Identify the Y-Value
The equation \( y = -3 \) indicates that the y-value for every point on this line is constantly -3.
03
Identify Points on the Line
Choose any x-values, such as \( x = 0 \), \( x = 1 \), and \( x = -1 \). For all of these x-values, the corresponding y-values must be -3. Thus, some points on the line are \((0, -3)\), \((1, -3)\), and \((-1, -3)\).
04
Draw the Line
Plot these points on the Cartesian plane. Since each point is horizontally aligned with \( y = -3 \), draw a straight horizontal line through them.
05
Final Graph Analysis
The resulting graph is a horizontal line that intersects the y-axis at -3 and runs parallel to the x-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Lines
In mathematics, a horizontal line is an interesting concept that surfaces when graphing linear equations. A horizontal line appears parallel to the x-axis and remains constant across all its points. When you have an equation like \( y = -3 \), it specifies that no matter what the x-coordinate is, the y-coordinate is always -3.
To better understand horizontal lines:
To better understand horizontal lines:
- The equation format is always \( y = c \) where \( c \) is a constant. In our example, this constant is -3.
- It means the line will not tilt or slope; it simply stretches left and right along the y-value of -3.
- For any x-value chosen, the y-value remains the same, thus forming a flat, straight line.
Coordinate Plane
The coordinate plane is a crucial tool in mathematics, especially when graphing equations. It's essentially a two-dimensional plane, formed by the intersection of a horizontal line called the x-axis and a vertical line called the y-axis.
Key features of the coordinate plane include:
Key features of the coordinate plane include:
- The intersection of the x-axis and y-axis is known as the origin, denoted by the point (0, 0).
- The plane is divided into four sections, or quadrants, which help in identifying the signs of the coordinates.
- Coordinates are written as ordered pairs \((x, y)\), allowing everyone to specify the exact location of a point.
Cartesian Plane
The Cartesian plane, often interchangeable with the term "coordinate plane," provides a graphical framework for plotting mathematical equations. This plane is named after the French mathematician Ren茅 Descartes, who significantly contributed to its development.
Why is the Cartesian plane important?
Why is the Cartesian plane important?
- It allows for the representation of algebraic equations in geometric terms, making them easier to visualize and understand.
- The plane enables us to plot points using a system of coordinates, making math more tangible.
- In a Cartesian plane, every point is defined precisely by an ordered pair \((x, y)\).
