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Chapter 1: Problem 32

In Exercises \(31-34,\) find the line's \(x\) - and \(y\) -intercepts and use this information to graph the line. $$ x+2 y=-4 $$

Short Answer

Expert verified
The x-intercept is (-4, 0) and the y-intercept is (0, -2).

Step by step solution

01

Identify Equation Form

We start with the linear equation given: \( x + 2y = -4 \). This is in standard form \( Ax + By = C \), where \( A = 1 \), \( B = 2 \), and \( C = -4 \).
02

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( x + 2y = -4 \). This results in \( x + 2(0) = -4 \), simplifying to \( x = -4 \). Thus, the x-intercept is \( (-4, 0) \).
03

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( x + 2y = -4 \). This results in \( 0 + 2y = -4 \), or \( 2y = -4 \). Solving for \( y \), we divide both sides by 2, yielding \( y = -2 \). Hence, the y-intercept is \( (0, -2) \).
04

Plot the Intercepts

Plot the intercepts found: the x-intercept \( (-4, 0) \) and the y-intercept \( (0, -2) \) on the Cartesian coordinate plane.
05

Draw the Line

Draw a straight line connecting the two plotted intercept points. This is the graph of the line represented by the equation \( x + 2y = -4 \).

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Key Concepts

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

Understanding X-Intercepts
An x-intercept is where a line crosses the x-axis. At this point, the value of y is always zero. You can find the x-intercept of an equation by substituting zero for y and solving for x. This gives you the precise point where the line meets the x-axis.
In our example equation, \( x + 2y = -4 \), setting \( y = 0 \) leads to \( x = -4 \). Thus, the x-intercept is \((-4, 0)\).

Understanding x-intercepts helps in graphing, because it gives one of the key locations through which the line passes. This is crucial for an accurate graph.
  • Set \( y = 0 \) in the equation to find the x-intercept.
  • Solve the resulting equation for x to get the intercept point.
  • Interpret the intercept as a coordinate \( (x, 0) \).
Understanding Y-Intercepts
A y-intercept is the point where a line crosses the y-axis. At this point, the value of x is always zero. To find the y-intercept, you need to set x to zero in the equation and solve for y. This tells you exactly where the line meets the y-axis.
Using the same example, \( x + 2y = -4 \), by setting \( x = 0 \), we find that \( y = -2 \). This makes the y-intercept \((0, -2)\).

Knowing the y-intercept is essential for graphing a line because it gives another vital point to accurately place the line on the graph.
  • Set \( x = 0 \) in the equation to determine the y-intercept.
  • Solve for y to find the corresponding intercept point.
  • Understand this intercept as a coordinate \( (0, y) \).
Graphing Lines Using Intercepts
Graphing a line using its x- and y-intercepts is straightforward and efficient. To graph a line, these intercepts are plotted on a coordinate plane to provide clear guidance on the line's path.
For the equation \( x + 2y = -4 \), you've already found two crucial points: the x-intercept \((-4, 0)\) and the y-intercept \((0, -2)\). These points help form the backbone of the line.

To graph:
  • First, plot the x-intercept \((-4, 0)\) on the graph.
  • Next, plot the y-intercept \((0, -2)\).
  • Finally, draw a straight line connecting these two points.
Using intercepts simplifies the graphing process, minimizing errors and ensuring accuracy.

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Most popular questions from this chapter

Graph the functions in Exercises \(29-48\) $$ y=\frac{1}{x^{2}}-1 $$

Consider the function \(y=\sqrt{(1 / x)-1}\) a. Can \(x\) be negative? b. \(\operatorname{Can} x=0 ?\) c. \(\operatorname{Can} x\) be greater than 1\(?\) d. What is the domain of the function?

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=x^{3} \quad \text { Left } 1, \text { down } 1 $$

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