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Chapter 1: Problem 2

Fill in the blanks. The point of intersection of the x- and y-axes is the ____ , and the two axes divide the coordinate plane into four parts called ____.

Short Answer

Expert verified
The point of intersection of the x- and y-axes is the origin, and the two axes divide the coordinate plane into four parts called quadrants.

Step by step solution

01

Recognizing the Intersection of X and Y-Axes

On a coordinate plane, the x and y axes intersect each other at a point where both x and y coordinates are equal to zero. This specific point is called the origin.
02

Naming the Divided Parts on the plane

When the x and y axes intersect, they divide the coordinate plane into four sections. Each of these sections is referred to as a quadrant.

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Key Concepts

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

Origin
In the realm of the coordinate plane, the term "origin" holds a special significance. This point, where the x-axis (horizontal) and y-axis (vertical) intersect, is a cornerstone of coordinate geometry.
In mathematical terms, the origin is denoted as \((0, 0)\).
It acts as a universal reference point, from which all other points are positioned and measured.
  • The origin is crucial for determining the position of a point in the two-dimensional space.
  • It serves as the starting point for both axes.
Whenever you locate a point on the coordinate plane, you begin at the origin and then move parallel to the axes according to the coordinates.
Axes
The axes on a coordinate plane are like the backbone of a map. You have two main axes: the x-axis and the y-axis.
The x-axis runs horizontally, while the y-axis stands vertically. These axes are central for graphing equations, plotting points, and shaping geometric figures.
  • X-axis: Represents horizontal direction. Increases to the right of the origin and decreases to the left.
  • Y-axis: Represents vertical direction. Values increase as you go upwards and decrease when moving downwards.
Both axes are infinite lines that extend in both directions, but they intersect at just one point, the origin. Together, they help define every point's position in a coordinate system using a pair of values \((x, y)\).
Quadrants
The coordinate plane is divided into four distinct regions called quadrants. These divisions result from the intersection of the x-axis and the y-axis at the origin.
Think of them as four separate neighborhoods on your coordinate map.
  • Quadrant I: Contains all points where both x and y are positive \((+, +)\).
  • Quadrant II: Here, x is negative, and y is positive \((- , +)\).
  • Quadrant III: Both x and y values are negative \((- , -)\).
  • Quadrant IV: Contains points where x is positive and y is negative \((+ , -)\).
Understanding these quadrants is vital for graphing equations and solving geometric problems. Each quadrant provides a different set of conditions for the coordinates within its boundaries.
Intersection Point
The intersection point on a coordinate plane is where the horizontal x-axis and the vertical y-axis cross each other. This point is inherently unique because it marks the center of the coordinate system.
It's not just any intersection; it serves a fundamental purpose in geometry and algebra.
  • Referred to as the "origin," the coordinates at this intersection are \((0, 0)\).
  • This intersection point is the reference for positioning all other points on the plane.
Through this single location, the coordinate plane is organized into a structured system. Every point can be described in relation to the intersection point, making it foundational in understanding spatial relationships on the plane.

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Most popular questions from this chapter

Bacteria Count The number \(N\) of bacteria in a refrigerated food is given by \(N(T)=10 T^{2}-20 T+600, \quad 2 \leq T \leq 20\) where \(T\) is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by \(T(t)=3 t+2, \quad 0 \leq t \leq 6\) where \(t\) is the time in hours. (a) Find the composition \((N \circ T)(t)\) and interpret its meaning in context. (b) Find the bacteria count after 0.5 hour. (c) Find the time when the bacteria count reaches 1500 .

The cost per unit in the production of an MP3 player is 60 dollars. The manufacturer charges 90 dollars per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by 0.15 dollars per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of 87 dollars per MP3 player for an order size of 120 ). (a) The table shows the profits \(P\) (in dollars) for various numbers of units ordered, \(x .\) Use the table to estimate the maximum profit. $$\begin{array}{|l|c|c|c|c|c|}\hline \text { Units, } x & 130 & 140 & 150 & 160 & 170 \\\\\hline \text { Profit, } P & 3315 & 3360 & 3375 & 3360 & 3315 \\\\\hline\end{array}$$ (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x ?\) (c) Given that \(P\) is a function of \(x,\) write the function and determine its domain. (Note: \(P=R-C\) where \(R\) is revenue and \(C\) is cost.)

\(g\) is related to one of the parent functions described in Section \(1.6 .\) (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g .\) (d) Use function notation to write \(g\) in terms of \(f\). $$g(x)=(x-8)^{2}$$

\(g\) is related to one of the parent functions described in Section \(1.6 .\) (a) Identify the parent function \(f\). (b) Describe the sequence of transformations from \(f\) to \(g .\) (c) Sketch the graph of \(g .\) (d) Use function notation to write \(g\) in terms of \(f\). $$g(x)=-|x|-2$$

The earnings per share for Big Lots, Inc. were \(\$ 1.89\) in 2008 and \(\$ 2.83\) in 2010 . Use the Midpoint Formula to estimate the earnings per share in 2009 . Assume that the earnings per share followed a linear pattern.
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