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Chapter 4: Problem 68

Graph each equation. \(y=-5\) (Section 3.2, Example 7)

Short Answer

Expert verified
The graph of the equation \(y=-5\) is a straight line parallel to the x-axis passing through points where y-coordinate is -5.

Step by step solution

01

Understand the equation

The equation given here is \(y=-5\). This is a linear equation where y is a constant. This means for all values of x, y will always be -5.
02

Plotting points on the graph

Since the y-coordinate remains -5 for all x-coordinates, we can pick any values for x and the corresponding y will always be -5. Let's pick few points, say x= -2, 0, 2, their corresponding y-values will be -5.
03

Drawing the graph

Join the points plotted on the Cartesian plane. The line passing through these points represents the graph of the given equation \(y=-5\). This is a horizontal line parallel to x-axis at y=-5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
A linear equation is a mathematical expression that forms a straight line when graphed on a coordinate plane. These equations take the general form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
In simpler terms, a linear equation involves variables where the highest power of the variable is one. This means the relationship between the variables shows as a line.
**Key points about linear equations:**
  • They can represent real-world relationships, like speed and time.
  • The slope \(m\) tells us the steepness and direction of the line.
  • The y-intercept \(b\) is where the line crosses the y-axis.
For our example, though, the equation \(y = -5\) doesn't show a slope part. It is essentially "flat" in terms of slope, which leads us to explore the next concept.
Cartesian Plane
The Cartesian plane is a two-dimensional grid where linear equations get visualized. It comprises two perpendicular axes: the horizontal x-axis and the vertical y-axis. These intersect at a point called the origin, labeled as (0, 0).
**Understanding the axes:**
  • The x-axis runs left to right. Typically, positive numbers are on the right, and negative numbers are on the left.
  • The y-axis runs up and down. Positive numbers are upwards, and negative numbers are downwards.
When plotting linear equations like \(y = -5\), we mark points on this grid. Each point corresponds to a pair of values \((x, y)\). Since the equation states \(y = -5\) for any \(x\), the points would spread along a horizontal line. Understanding how these axes work helps us graph the equation efficiently.
Horizontal Lines
Horizontal lines on a graph are unique and straightforward. They occur when the y-value remains constant, regardless of the x-value. Our example, \(y = -5\), forms such a line.
**Characteristics of horizontal lines:**
  • The line's slope is zero, meaning it's completely flat.
  • These lines run parallel to the x-axis.
  • They intersect the y-axis at the point where the y-value is constant, here at \(y = -5\).
To graph a horizontal line, once you know the constant y-value, use any x-values. Plot the respective points, and they should align horizontally across the plane. Understanding horizontal lines simplify tasks like predicting the behavior of such equations and identifying them quickly on a graph.

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