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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

A roofing contractor purchases a shingle delivery truck with a shingle elevator for 42,000 dollar. The vehicle requires an average expenditure of 9.50 dollar per hour for fuel and maintenance, and the operator is paid 11.50 dollar per hour. (a) Write a linear equation giving the total cost \(C\) of operating this equipment for \(t\) hours. (Include the purchase cost of the equipment.) (b) Assuming that customers are charged 45 dollar per hour of machine use, write an equation for the revenue \(R\) derived from \(t\) hours of use. (c) Use the formula for profit \(P=R-C\) to write an equation for the profit derived from \(t\) hours of use. (d) Use the result of part (c) to find the break-even point - that is, the number of hours this equipment must be used to yield a profit of 0 dollars.

a) find the inverse function of \(f\) (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1},\) and (d) state the domains and ranges of \(f\) and \(f^{-1}\). $$f(x)=3 x+1$$

Finding a Mathematical Model In Exercises \(41-50\), find a mathematical model for the verbal statement. Newton's Law of Universal Gravitation: The gravitational attraction \(F\) between two objects of masses \(m_{1}\) and \(m_{2}\) is jointly proportional to the masses and inversely proportional to the square of the distance \(r\) between the objects.

The table shows the life expectancies of a child (at birth) in the United States for selected years from 1930 through \(2000\) . $$\begin{array}{|c|c|}\hline \text { Year } & \text { Life Expectancy, \(y\) } \\\\\hline 1930 & 59.7 \\\1940 & 62.9 \\\1950 & 68.2 \\\1960 & 69.7 \\\1970 & 70.8 \\\1980 & 73.7 \\\1990 & 75.4 \\\2000 & 76.8 \\\\\hline\end{array}$$ A model for the life expectancy during this period is $$y=-0.002 t^{2}+0.50 t+46.6, \quad 30 \leq t \leq 100$$ where \(y\) represents the life expectancy and \(t\) is the time in years, with \(t=30\) corresponding to 1930. (a) Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain. (b) Determine the life expectancy in 1990 both graphically and algebraically. (c) Use the graph to determine the year when life expectancy was approximately \(76.0 .\) Verify your answer algebraically. (d) One projection for the life expectancy of a child born in 2015 is \(78.9 .\) How does this compare with the projection given by the model? (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explain.

For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\) (b) Use the function in part (a) to complete the table. What can you conclude? $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline n & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\\\\hline R(n) & & & & & & & \\\\\hline\end{array}$$
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