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Chapter 2: Problem 57

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ y+2=0 $$

Short Answer

Expert verified
y-intercept: (0, -2), No x-intercept, Horizontal line at y = -2.

Step by step solution

01

Isolate the variable for y-intercept

Given the equation: \[ y + 2 = 0 \]Subtract 2 from both sides to isolate y:\[ y = -2 \]This tells us that the y-intercept is at the point (0, -2).
02

Determine the x-intercept

To find the x-intercept, we need to set y to 0 in the equation. However, the equation \[ y = -2 \]is always equal to -2 and does not depend on x. Thus, there is no x-intercept since the value of y does not allow it to cross the x-axis.
03

Graph the equation

To graph the equation \[ y = -2 \]plot a horizontal line through the point (0, -2). This line extends infinitely in the positive and negative x-directions but remains fixed at y = -2.

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Key Concepts

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

x-intercept
The x-intercept is where a graph crosses the x-axis. To find the x-intercept of an equation, we set y to 0 and solve for x. This gives us the point where the line touches the x-axis, meaning y is zero at this point.
In the given equation, \[y + 2 = 0 \], we isolate y and get \[ y = -2 \]. This equation tells us that y is always -2 and does not change with x.
So, we cannot find an x-intercept because the line never touches the x-axis. The graph is a horizontal line at y = -2.
y-intercept
The y-intercept is where a graph crosses the y-axis. This occurs when x = 0. To find the y-intercept, we set x to 0 and solve for y.
In our given equation, \[y + 2 = 0 \], we subtract 2 from both sides to isolate y:\[ y = -2 \]. This tells us the y-intercept is at the point \((0, -2)\).
On a graph, this means that the line will touch the y-axis at this point. For this equation, the line is horizontal and passes through \((0, -2)\).
graphing linear equations
Graphing linear equations is an important skill in algebra. It allows us to visualize the relationship between x and y.To graph a linear equation like \[ y + 2 = 0 \], follow these steps:
  1. Simplify the equation to its standard form, which in this case means isolating y.
  2. Identify the y-intercept. Here, y = -2, so the y-intercept is \((0, -2)\).
  3. Plot the y-intercept on the graph.
  4. Recognize the line's characteristics. Since y is always -2, plot a horizontal line through this point.

By plotting the y-intercept and understanding that the equation represents a horizontal line, you can easily draw the graph.
Remember, linear equations can take many forms, but the process of identifying intercepts and plotting remains similar. When x is involved, finding both intercepts helps in drawing the correct line.

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Most popular questions from this chapter

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ 4 x-3 y=8 $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(3 x=y\) and \(2 y-6 x=5\)

Graph each line passing through the given point and having the given slope. (0,0)\(; m=\frac{5}{3}\)

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ x-4 y=8 $$

Graph each line passing through the given point and having the given slope. (-1,-2)\(; m=3\)
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