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Chapter 4: Problem 29

Graph each equation by finding the intercepts and at least one other point. $$y+3=0$$

Short Answer

Expert verified
To graph the equation \(y + 3 = 0\), find the intercepts and at least one other point: 1. There is no x-intercept for the given equation. 2. The y-intercept is (0, -3). 3. Another point on the graph is (1, -3). The graph is a horizontal line at \(y = -3\).

Step by step solution

01

Find the x-intercept

The x-intercept is the point where the graph intersects the x-axis, at which the y-coordinate is equal to 0. We'll substitute \(y=0\) into the given equation to find the x-intercept. \(0 + 3 = 0\) There is no value of x in the equation, so there's no x-intercept for the given equation.
02

Find the y-intercept

The y-intercept is the point where the graph intersects the y-axis, at which the x-coordinate is equal to 0. The equation is already in the form \(y + 3 = 0\), so the y-intercept is the point where \(y = -3\). Thus, the y-intercept is (0, -3).
03

Find at least one other point on the graph

Since the given equation is a horizontal line, we can pick any x-coordinate and the y-coordinate will remain -3. Let's choose x = 1 to find another point on the graph. Point: (1, -3)
04

Graph the equation

Now that we have the y-intercept and another point on the graph, we can plot these points on the coordinate plane and draw a horizontal line through them: 1. Plot the y-intercept (0, -3) on the y-axis. 2. Plot the additional point (1, -3) on the graph. 3. Draw a horizontal line through these points, representing the equation \(y + 3 = 0\), which is a horizontal line at \(y = -3\). The graph of the equation is a horizontal line at \(y = -3\).

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Key Concepts

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

Intercepts in Graphing
When graphing equations, one of the simplest ways to begin is by finding the intercepts. Intercepts are the points where the graph of the equation crosses the axes of a coordinate plane. There are two types of intercepts:
  • X-intercept: This is the point where the graph crosses the x-axis. At this point, the y-coordinate is always zero. However, not every line will have an x-intercept, particularly horizontal lines where the y-value doesn鈥檛 touch the x-axis.
  • Y-intercept: This is the point where the graph crosses the y-axis. At this point, the x-coordinate is always zero. The y-intercept is often easy to spot when the equation is in a simple format.
To find intercepts, substitute the appropriate coordinate to zero and solve for the other variable. Understanding intercepts is crucial as they provide a foundation for plotting other points on the graph.
Horizontal Lines
Horizontal lines have special characteristics in graphing. These lines are parallel to the x-axis and have the same y-coordinate at every point. The equation of a horizontal line is usually represented in the form of \( y = c \) where \( c \) is a constant number. This means every point on the line has a y-coordinate of \( c \), and the x can vary freely.

Some features of horizontal lines include:
  • No x-intercept: Because the line is never vertical, it does not intersect the x-axis.
  • Consistent y-coordinate: Every point along the line will have the same y-value, making it easy to plot the line if you know the y-intercept.
For example, in the equation \( y + 3 = 0 \), simplifying gives \( y = -3 \). Every point on this line has a y-coordinate of \(-3\). Therefore, the line is horizontal and remains consistent across various x-values.
Y-intercept
The y-intercept is a key concept when it comes to graphing鈥攁 starting point on the y-axis. It is where the graph of an equation intersects the y-axis and can be found by setting the x-value to zero and solving for y.

A y-intercept can tell you:
  • Location on y-axis: It provides a quick location of the graph鈥檚 starting point along the y-axis when x is zero.
  • Behavior of the graph: For vertical lines and other more complex graphs, knowing the y-intercept helps in understanding the initial slope or direction.
In the equation \( y + 3 = 0 \), when solving for y-intercept, you set x to 0, and you find that \( y = -3 \). This means the y-intercept is at the point \( (0, -3) \). Understanding the y-intercept makes it easier to graph the rest of your equation.

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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 \(y=3\) containing \((2,1)\)

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=-1\) and contains \((8,-8)\)

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=-\frac{5}{9-3 x}$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(y\) -intercept \((0,6)\) and \(m=7\)

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=2\) and \(y\) -intercept \((0,-11)\)
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