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Chapter 4: Problem 27

Graph each equation by finding the intercepts and at least one other point. $$y=0$$

Short Answer

Expert verified
The x-intercept and y-intercept for the equation \(y=0\) is at the point (0, 0). Another point on the line is (2, 0). To graph the equation, plot these points on the coordinate plane and connect them with a straight horizontal line. This line represents the equation \(y = 0\).

Step by step solution

01

Finding the intercepts

To find the x-intercept and y-intercept, we will set x = 0 and y = 0 respectively and solve for the other variable. 1. x-intercept: Set y = 0, but the equation is already y = 0, so the x-intercept is any value of x. We can choose x = 0 for simplicity. Thus, the x-intercept is at the point (0, 0). 2. y-intercept: Set x = 0, but the equation is already y = 0. The point (0, 0) is also the y-intercept point.
02

Finding an additional point

Let's find another point on the line by choosing a value for x. Let x = 2, substituting this value into the equation y = 0, we have: y = 0 Thus, the point (2, 0) is also on the line with the equation y = 0.
03

Graphing the equation

Now that we have found all the necessary points, we can graph the equation y = 0 with these points: (0, 0), and (2, 0). After plotting these points on the coordinate plane, connect the points with a straight line. The line will be a horizontal line that passes through all points with y-coordinate of 0.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a graph is the point where the line crosses the x-axis. In simpler terms, it's where the value of y is zero. This means that at the x-intercept, the equation of the graph has the form of (x, 0). This concept emphasizes understanding how lines behave at this critical point.
To find the x-intercept for a linear equation like \(y = 0\), you set y to zero and solve for x. In our exercise, since the equation is \(y = 0\), this means y is already zero regardless of x. All points along the x-axis are solutions, but we commonly choose (0,0) for clarity when graphing.
Exploring the Y-Intercept
Just as the x-intercept relates to the x-axis, the y-intercept is where the line crosses the y-axis. Here, the x-value is zero, and the focus is on the y-value. This is crucial when plotting root lines or determining their slope and starting point.
For the equation \(y = 0\), we set x equal to zero and solve for y. Because the equation states \(y = 0\), again, all points along the x-axis apply, as the graph just meets the y-intercept exactly at the point (0,0). This gives a single solution where it touches the origin.
Characterizing the Horizontal Line
Horizontal lines are unique in that they have a constant y-value for all x-values. In our specific case, the equation \(y = 0\) implies that the line is purely horizontal, positioned exactly on the x-axis.
A characteristic of such lines is their slope, which is 0. This means there's no 'rise' over the run, signifying a flat, even line. Often, horizontal lines are easy to graph due to their simplicity - every point on the line has the same y-coordinate.
  • Flat and constant: No upward or downward movement.
  • Easy identification: Always parallel to the x-axis.
Navigating the Coordinate Plane
A coordinate plane, also known as a Cartesian plane, is a two-dimensional surface defined by a north-south vertical line (y-axis) and an east-west horizontal line (x-axis). This grid allows us to plot points, lines, and curves using pairs of numbers.
Each point on this plane is determined by an x-coordinate (horizontal position) and a y-coordinate (vertical position). When you graph equations, you are essentially marking these points in order to find the shape or the path described by the equation. In the exercise, points like (0, 0) and (2, 0) are specific coordinates that help illustrate that the graph of \(y = 0\) will be a straight horizontal line along the x-axis.

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Most popular questions from this chapter

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(21 x-6 y=2\) containing \((4,-1)\)

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