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Chapter 1: Problem 75

Sketch the graph of the equation. Use a graphing utility to verify your result. $$ y=-2 $$

Short Answer

Expert verified
The solution is a horizontal line at \(y = -2\) which extends indefinitely on both sides of the x-axis.

Step by step solution

01

Understanding the equation

Given the equation \(y = -2\), it can be seen that there is no dependence on \(x\) meaning that the \(y\) value will always remain -2, regardless of the \(x\) value. This represents a horizontal line in the y-coordinate plane.
02

Plot the line

Since \(y\) is always equal to -2, the horizontal line is located at \(y = -2\) on the coordinate plane. Draw a horizontal line through the point \((0, -2)\). This line extends indefinitely in both the positive and negative x-directions.
03

Verification using a graphing utility

To confirm the sketch, use a graphing utility. Input the equation \(y = -2\) into the graphing tool. The resulting graph should coincide with the earlier sketch of a horizontal line crossing the y-axis at -2.

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Key Concepts

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

Understanding the Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It's made up of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin (0,0).

Each point on the plane is defined by a pair of numerical coordinates: \( (x, y) \), where \( x \) represents the horizontal position and \( y \) the vertical position. By understanding this concept, sketching graphs like a horizontal line becomes a matter of knowing one of those coordinates. In our case, for any given \( x \), the \( y \) coordinate is always -2, meaning all points of the line will have -2 as their second coordinate.
Graphing Utility Essentials
A graphing utility is a digital tool that helps us create visual representations of algebraic equations. It saves time and provides a precise graph that is often difficult to achieve on paper, especially for complex functions.

To use a graphing utility effectively, you need to input the equation you are working with and adjust the viewing window to see the relevant parts of the graph. For the exercise \( y = -2 \), entering this simple horizontal line equation should result in a straight line graph where the line passes through all points that have a \( y \) value of -2.

Tip for Accuracy:

  • Always check your graph's scale to ensure you're interpreting it correctly.
  • Use the utility's tracing function to see exact values of points on the graph.
Horizontal Line Equation
The equation for a horizontal line is deceptively simple: it has the form \( y = k \) where \( k \) is a constant. This tells us that the value of \( y \) does not change no matter what \( x \) is.

Therefore, if we're given \( y = -2 \), this represents a line where every point has the \( y \) value of -2. It is parallel to the x-axis and intersects the y-axis at the point where \( y \) is -2. The concept of the horizontal line equation is critical as it sets the foundation for understanding more complex linear equations and how they graph on a coordinate plane.

Visualization on the Plane:

  • Imagine walking along the line: your altitude doesn't change, which is exactly what a constant \( y \) value means in the context of a horizontal line.
  • Conversely, a vertical line has the equation \( x = k \) and would mean your horizontal position remains constant while you move up or down.

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Most popular questions from this chapter

Use a graphing utility to estimate the limit (if it exists). \(\lim _{x \rightarrow 1} \frac{x^{2}+6 x-7}{x^{3}-x^{2}+2 x-2}\)

Find the limit. \(\lim _{x \rightarrow 2}\left(-x^{2}+x-2\right)\)

Biology The gestation period of rabbits is about 29 to 35 days. Therefore, the population of a form (rabbits' home) can increase dramatically in a short period of time. The table gives the population of a form, where \(t\) is the time in months and \(N\) is the rabbit population. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline N & {2} & {8} & {10} & {14} & {10} & {15} & {12} \\\ \hline\end{array} $$ Graph the population as a function of time. Find any points of discontinuity in the function. Explain your reasoning.

MAKE A DECISION: REVENUE For groups of 80 or more people, a charter bus company determines the rate \(r\) (in dollars per person) according to the formula \(r=15-0.05(n-80), \quad n \geq 80\) where \(n\) is the number of people. (a) Express the revenue \(R\) for the bus company as a function of \(n .\) (b) Complete the table. (c) Criticize the formula for the rate. Would you use this formula? Explain your reasoning.

Find the limit (if it exists). \(\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}\)
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