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Chapter 3: Problem 94

Explain in your own words why the graph of \(x=-4\) is a vertical line.

Short Answer

Expert verified
The graph of \(x = -4\) is a vertical line because x is always -4 regardless of the value of y.

Step by step solution

01

- Understand the Equation

The given equation is in the form of 'x equals a constant'. Here, the equation is given as \(x = -4\).
02

- Identify the Key Characteristic

For any equation of the form \(x = c\), where c is a constant, the value of x is always fixed. In this case, no matter what the value of y is, x will always be -4.
03

- Plotting the Points

To visualize this, plot several points on the graph where x is always -4. For example, (-4, 1), (-4, 2), (-4, -3), and so on. Notice that these points align vertically.
04

- Drawing the Line

Since all plotted points have the same x-coordinate and different y-coordinates, connect these points to form a vertical line through x = -4.
05

Conclusion

The graph of \(x = -4\) is a vertical line because the x-coordinate is fixed at -4 for all values of y.

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Key Concepts

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

Vertical Line Graph
The graph of the equation \(x = -4\) is a vertical line. This is because the equation tells us that the value of \(x\) does not change; it remains constant at -4 for all points along the line. A vertical line goes straight up and down and does not tilt.
Understanding vertical lines is crucial in graphing. Unlike typical lines that slant at an angle or horizontal lines that run left to right, vertical lines remain completely upright.
Key aspects include:
  • A vertical line has an undefined slope.
  • The equation is always in the form \(x = c\), where \(c\) is a constant value.

Knowing these points helps in correctly identifying and graphing vertical lines, which is important for further studies in coordinate geometry.
Constant x-coordinate
In the equation \(x = -4\), the \(x\)-coordinate is constant. This means that for any value of \(y\), the corresponding \(x\)-value will always be -4.
This concept is key to understanding why the graph of \(x = -4\) is vertical. It tells us:
  • No matter how much \(y\) changes, \(x\) does not.
  • The value of \(x\) is fixed at -4.
This characteristic leads to every point on the line sharing the same \(x\)-coordinate, like (-4, 1), (-4, -2), etc.
Recognizing that the \(x\)-coordinate remains fixed helps in plotting and graphing these points, reinforcing the concept of vertical lines in coordinate geometry.
Plotting Points
Plotting points is the process of marking locations on a graph based on their coordinates. For the equation \(x = -4\), you plot several points where the \(x\)-coordinate is always -4.
Examples include points like (-4, 1), (-4, 2), (-4, -3). Although the \(y\)-values differ, the \(x\) value remains the same, which causes these points to line up vertically.
Here's a step-by-step approach:
  • Choose different values for \(y\).
  • For each \(y\) value, the corresponding \(x\) value will always be -4.
  • Mark each point on the graph.

Connecting these points will reveal the vertical line, further illustrating the concept of a constant \(x\)-coordinate.

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Most popular questions from this chapter

To prepare for Section \(3.5,\) review subtraction and order of operations. Simplify. $$ \frac{-4-8}{7-(-2)} $$

The greatest integer function \(f(x)=[x]\) is defined as follows: \([x]\) is the greatest integer that is less than or equal to \(x .\) For example, if \(x=3.74,\) then \([x]=3\) and if \(x=-0.98,\) then \([x]=-1 .\) Graph the greatest integer function for \(-5 \leq x \leq 5 .\) (The notation \(f(x)=\operatorname{int}(x),\) used in many graphing calculators, is often found in the MATH NUM submenu.)

Abby asserts that for a function described by a set of ordered pairs, the range of the function will always have the same number of elements as there are ordered pairs. Is she correct? Why or why not?

If \(f(x)=0.1 x-0.5,\) for what input is the output \(-3 ?\)

Graph the equations $$\begin{array}{ll}{y_{1}=1.4 x+2,} & {y_{2}=0.6 x+2} \\\\{y_{3}=1.4 x+5,} & {\text { and } \quad y_{4}=0.6 x+5}\end{array}$$ using a graphing calculator. If possible, use the Simultaneous mode so that you cannot tell which equation is being graphed first. Then decide which line corresponds to each equation.
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