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Chapter 3: Problem 49

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

Short Answer

Expert verified
The y-intercept is (0, -2). There is no x-intercept. The graph is a horizontal line at y = -2.

Step by step solution

01

Find the y-intercept

The y-intercept occurs when the value of x is 0. In the given equation, set x to 0 and solve for y. The equation is \[ y + 2 = 0 \] To isolate y, subtract 2 from both sides: \[ y = -2 \] Therefore, the y-intercept is (0, -2).
02

Find the x-intercept

The x-intercept occurs when the value of y is 0. In the given equation, substitute y with 0: \[ 0 + 2 = 0 \]However, solving this does not provide a valid x value as it leads to a false statement. Therefore, this equation does not have an x-intercept.
03

Graph the equation

The equation \[ y + 2 = 0 \], which simplifies to \[ y = -2 \], represents a horizontal line. On a graph, draw a horizontal line passing through the y-coordinate of -2, ensuring it extends uniformly across all values of x.

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Key Concepts

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

Finding the Y-Intercept
To find the y-intercept of an equation, you need to determine what the value of y is when x is equal to zero. This concept is straightforward but crucial for graphing.

For the equation \(
y + 2 = 0 \), set x to 0. Since there is no x term in this particular equation, we directly solve for y.
  • Subtract 2 from both sides to isolate y: \[ y = -2 \]
  • This tells us that the y-intercept is at \((0, -2)\).


So, anytime you want to find the y-intercept, just substitute x with zero and solve for y.
Finding the X-Intercept
Finding the x-intercept is similar in concept to finding the y-intercept but involves setting y to zero. The x-intercept is where the graph crosses the x-axis.

Take the equation \(
y + 2 = 0 \), and set y to 0:
  • \[ 0 + 2 = 0 \]
  • However, simplifying this equation leads to \[ 2 = 0 \], which is a contradiction.
  • This equation does not yield any valid x value.

Therefore, there is no x-intercept for this particular equation. Remember that some equations might not have an x-intercept, especially horizontal lines like the one we are dealing with here.
Graphing Equations
Graphing an equation helps visualize its behavior on the coordinate plane. Let's use our equation: \[ y + 2 = 0 \]
  • First, simplify it to its easiest form: \[ y = -2 \].
  • This is a horizontal line.
  • A horizontal line maintains the same y value (\

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Most popular questions from this chapter

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. There is a \(\$ 30\) fee to rent a chain saw, plus \(\$ 6\) per day. Let \(x\) represent the number of days the saw is rented and \(y\) represent the charge to the user in dollars. If the total charge is \(\$ 138,\) for how many days is the saw rented?

Write an equation in the form \(y=m x\) for each situation. Then give the three ordered pairs associated with the equation for x-values 0,5, and 10. See Example 7(a). \(x\) represents the number of credit hours taken at Kirkwood Community College at \(\$ 111\) per credit hour, and \(y\) represents the total tuition paid for the credit hours (in dollars).

Solve each equation for \(y\). $$ 4 x-y=10 $$

Find the midpoint of each segment with the given endpoints. $$ (-8,4) \text { and }(-2,-6) $$

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x-3 y=0 $$
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