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Chapter 1: Problem 7

In Exercises 7-10, plot the points in the Cartesian plane. \( (-4, 2) \), \( (-3, -6) \), \( (0, 5) \), \( (1, -4) \)

Short Answer

Expert verified
The points have been successfully plotted on the Cartesian plane.

Step by step solution

01

Understand and set up the Cartesian Plane

Before plotting the points, it is vital to understand how the Cartesian plane works. A Cartesian plane has two axes: the x-axis (horizontal) and the y-axis (vertical). The point where these two axes intersect is called the origin (0, 0).
02

Plot the first point (-4, 2)

Start at the origin, move 4 units to the left (because of -4 as the x-coordinate) and 2 units upward (because of 2 as the y-coordinate) to plot the first point.
03

Plot the second point (-3, -6)

Again start at the origin, move 3 units to the left (due to -3 as the x-coordinate) and 6 units downward (due to -6 as the y-coordinate) to mark the second point.
04

Plot the third point (0, 5)

For the third point, stay at the origin for the x-coordinate (0 on the x-axis) and move 5 units upward for the y-coordinate to mark the point.
05

Plot the fourth point (1, -4)

Finally, from the origin, move 1 unit to the right (1 on the x-axis) and 4 units downward (-4 on the y-axis) to mark the fourth and final point.

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Key Concepts

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

Cartesian Coordinates
The concept of Cartesian coordinates is fundamental in mathematics. They allow us to identify specific points on a plane using a pair of numerical values.

These values correspond to the x (horizontal) and y (vertical) axes on the Cartesian plane. The point where these axes intersect is called the origin, denoted as (0, 0).

In the Cartesian coordinate system, each point is represented by a pair
  • The first number: x-coordinate, indicates the horizontal position.
  • The second number: y-coordinate, indicates the vertical position.

For example, the point \((-4, 2)\) has an x-coordinate of -4, meaning it is 4 units to the left of the y-axis, and a y-coordinate of 2, meaning it is 2 units above the x-axis.
Understanding Cartesian coordinates is key to navigating the Cartesian plane.
Plotting Points
Plotting points accurately is an essential skill in using the Cartesian plane.

To plot a point like \((-4, 2)\), you must start at the origin, move horizontally to the left by 4 units, and then move vertically up by 2 units.

This method can then be repeated for each coordinate you want to plot. Here's a simple guide:
  • For negative x-values, move left from the origin; for positive, move right.
  • For negative y-values, move down from the coordinate on the x-axis; for positive, move up.

Visualizing these movements on the Cartesian plane is like following directions on a map. For the point \((0, 5)\), you remain directly above the origin, moving only vertically.

This process becomes intuitive with practice, aiding in tasks like graphing and data analysis.
Coordinate System
The coordinate system is a structured layout that offers a way to locate points on a plane. It consists of two axes:
  • x-axis: running horizontally
  • y-axis: running vertically

The planes are divided into four quadrants:
  • Quadrant I: positive x and positive y
  • Quadrant II: negative x and positive y
  • Quadrant III: negative x and negative y
  • Quadrant IV: positive x and negative y

Each quadrant helps find the general position of a point. For example, the point \((-3, -6)\) is in Quadrant III.

Understanding the coordinate system is essential, enabling precise plotting and visualization of mathematical functions and real-world data.

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Most popular questions from this chapter

SPORTS The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are listed below. (Source: International Olympic Committee) 1920 146.6 1924 151.3 1928 155.3 1932 162.3 1936 165.6 1948 173.2 1952 180.5 1956 184.9 1960 194.2 1964 200.1 1968 212.5 1972 211.3 1976 221.5 1980 218.7 1984 218.5 1988 225.8 1992 213.7 1996 227.7 2000 227.3 2004 229.3 2008 225.8 (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920. (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the \(regression\) feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men's discus throw in the year 2012.

In Exercises 49-58, find a mathematical model for the verbal statement. \(A\) varies directly as the square of \(r\).

In Exercises 87-92, use the functions given by \(f(x) = \frac{1}{8}x - 3\) and \(g(x) = x^3\) to find the indicated value or function. \((f^{-1} \circ g^{-1})(1)\)

In Exercises 27-30, use the given value of \(k\) to complete the table for the inverse variation model \(y = \frac{k}{x^2}\) Plot the points on a rectangular coordinate system. \(k = 2\)

In Exercises 87-92, use the functions given by \(f(x) = \frac{1}{8}x - 3\) and \(g(x) = x^3\) to find the indicated value or function. \((g^{-1} \circ f^{-1})(-3)\)
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