Chapter 2 Integers And Rational Numbers
Section 2.4: Represent Rational Numbers On the Coordinate Plane
Page 85 Exercise 1 Answer
The point A is (3,5). Since point B has the same x-coordinate and the opposite y-coordinate, the x-coordinate is 3 and the y-coordinate is the opposite of 5, which is −5. The coordinates of the point B are (3,−5).
Result
B(3, -5)
Page 85 Exercise 1 Answer
If two points have the same x-coordinates and the opposite y-coordinates, they form mirror images of each other across the x-axis. For example, A(3,1) and B(3,−1).
Result
If two points have the same x-coordinates and the opposite y-coordinates, they form mirror images of each other across the x-axis.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 86 Exercise 1 Answer
To graph the point P(−2,−3) start at the origin (0,0). The x-coordinate is negative, −2, so move 2 units to the left. Then use the y-coordinate, which is −3, so move 3 units down.
Both x-coordinates and y-coordinates of all the points in quadrant I are positive, and both coordinates of all the points in quadrant III are negative. All the points in quadrant II have negative x-coordinates and positive y-coordinates, and all the points in quadrant IV have positive x-coordinates and negative y-coordinate.
Result
origin (0,0)
move 2 units to the left
move 3 units down
Both x-coordinates and y-coordinates of all the points in quadrant I are positive, and both coordinates of all the points in quadrant III are negative. All the points in quadrant II have negative x-coordinates and positive y-coordinates, and all the points in quadrant IV have positive x-coordinates and negative y-coordinate.
Page 87 Exercise 2 Answer
To find the landmark which is located on the map at (2, \(\frac{1}{4}\)) we first find 2 on the x-axis and \(\frac{1}{4}\) on the y-axis. We follow the grid lines, from 2 on x-axis and from \(\frac{1}{4}\) on y-axis, to where they meet. At that spot is the FBI Building.
Notice that instead of looking for \(\frac{1}{4}\) on y-axis, we could have looked for 0.25 on the y-axis since \(\frac{1}{4}\) = 0.25.
Result
At (2, \(\frac{1}{4}\)) on the map is the FBI Building.
Page 87 Exercise 3 Answer
Since B is a reflection of point A across the x-axis they differ only in the sign of the y-coordinates. Their x-coordinates are the same, so the x-coordinate of point B is -3. The y-coordinate of point B has a different sign than that of point A, so it is -5.
Result
The coordinates of point B are (−3,−5).
Page 88 Exercise 1 Answer
Evaluating expression with fractions is like evaluating an expression with whole numbers because we still use the same operations to evaluate the expression. The difference is that when dividing by a fraction, we must rewrite it as multiplication, which is something we don’t do when dividing by a whole number.
Page 88 Exercise 2 Answer
The y-coordinate of any point that lies on the x-axis is zero.
Since x-axis and y-axis are perpendicular to each other and intersect at (0,0), there will be no change in the y-coordinate no matter where you are on the x-axis. It will always be 0.
Result
It’s y-coordinate is zero.
Page 88 Exercise 3 Answer
Points (4,5) and (−4,5) differ only in the sign of the x-coordinate, thus they are reflections of each other across the y-axis.
Result
They are reflections of each other across the y-axis.
Page 88 Exercise 4 Answer
Since the coordinates of the landmark are (8,−10), the x-coordinate is positive and the y-coordinate is negative. We know that all points in Quadrant IV have a positive x-coordinate and a negative y-coordinate, so the point (8,−10) must be in Quadrant IV.
Result
Quadrant IV since the x-coordinate is positive and the y-coordinate is negative.
Page 88 Exercise 5 Answer
To graph the point A(−4,1), we find −4 on the x-axis and 1 on the y-axis. Follow the grid lines to where they meet, that is the point A.
Result
Find −4 on the x-axis and 1 on the y-axis. Follow the grid lines to where they meet, that is the point A.
Page 88 Exercise 6 Answer
To graph the point B(4,3), we find 4 on the x-axis and 3 on the y-axis. Follow the grid lines to where they meet, that is the point B.
Result
Find 4 on the x-axis and 3 on the y-axis. Follow the grid lines to where they meet, that is the point B.
Page 88 Exercise 7 Answer
To graph the point C(0,−2), we find 0 on the x-axis and −2 on the y-axis. Follow the grid lines to where they meet, that is the point C.
Result
Find 0 on the x-axis and −2 on the y-axis. Follow the grid lines to where they meet, that is the point C.
Page 88 Exercise 8 Answer
Follow the grid lines from the point P to the x-axis to find the x-coordinate, which is 3, and follow the grid lines to the y-axis to find the y-coordinate, which is −2.
Result
The coordinates of the point P are (3,−2).
Page 88 Exercise 9 Answer
To find the ordered pair of the White House follow the grid line to the x-axis to find the x-coordinate, which is 0.5 or written as a fraction, \(\frac{1}{2}\). Follow the grid line to the y-axis to find the y-coordinate, which is 0.75 or written as a fraction, \(\frac{3}{4}\).
Result
The ordered pair of the location of the White House is (\(\frac{1}{2}\), \(\frac{3}{4}\)).
Page 88 Exercise 10 Answer
To find the ordered pair of the Lincoln Memorial follow the grid line to the x-axis to find the x-coordinate, which is −1.25 or written as a fraction, –\(\frac{5}{4}\). Follow the grid line to the y-axis to find the y-coordinate, which is −0.75 or written as a fraction, –\(\frac{3}{4}\).
Result
The ordered pair of the location of the White House is (-\(\frac{5}{4}\), –\(\frac{3}{4}\)).
Page 88 Exercise 11 Answer
Since the y-coordinate is 0 the point is on the x-axis. Find 0.5 on the x-axis and that is the point which marks the landmark – the Ellipse.
Result
The ordered pair (0.5,0) marks the Ellipse.
Page 88 Exercise 12 Answer
Find \(\frac{3}{4}\) on the x-axis and –\(\frac{1}{2}\) on the y-axis. Follow the grid lines from \(\frac{3}{4}\) on the x-axis and –\(\frac{1}{2}\) on the y-axis to where they meet, that place marks the landmark – the Washington Monument.
Result
The ordered pair (\(\frac{3}{4}\), –\(\frac{1}{2}\)) marks the Washington Monument.
Page 89 Exercise 13 Answer
Follow the grid lines from 1 on the x-axis and from −1 on the y-axis to where they meet and mark the point A.
Page 89 Exercise 14 Answer
Follow the grid lines from 4 on the x-axis and from 3 on the y-axis to where they meet and mark the point B.
Page 89 Exercise 15 Answer
Follow the grid lines from −4 on x-axis and from 3 on y-axis to where they meet and mark that point C.
Page 89 Exercise 16 Answer
Follow the grid lines from 5 on x-axis and from −2 on y-axis to where they meet and mark that point D.
Page 89 Exercise 17 Answer
Follow the grid lines from −2.5 on x-axis and from 1.5 on y-axis to where they meet and mark that point E.
Page 89 Exercise 18 Answer
Follow the grid lines from 2 on x-axis and from 1.5 on y-axis to where they meet and mark that point F.
Page 89 Exercise 19 Answer
Follow the grid lines from −2 on x-axis and from –\(1 \frac{1}{2}\) on y-axis to where they meet and mark that point G.
Page 89 Exercise 20 Answer
Follow the grid lines from \(1 \frac{1}{2}\) on x-axis and from −1 on y-axis to where they meet and mark that point H.
Page 89 Exercise 21 Answer
Follow the grid line to where it crosses the x-axis to find the x-coordinate, −8. Follow the grid line to where it crosses the y-axis to find the y-coordinate, 3. The ordered pair for the point P is (−8,3).
Result
The ordered pair for the point P is (−8,3).
Page 89 Exercise 22 Answer
Follow the grid line to where it crosses the x-axis to find the x-coordinate, 5. Follow the grid line to where it crosses the y-axis to find the y-coordinate, −3. The ordered pair for the point Q is (5,−3).
Result
The ordered pair for the point Q is (5,−3).
Page 89 Exercise 23 Answer
Follow the grid line to where it crosses the x-axis to find the x-coordinate, −8. The point R is on the x-axis so the y-coordinate is zero. The ordered pair for the point R is (−8,0).
Result
The ordered pair for the point R is (−8,0).
Page 89 Exercise 24 Answer
Follow the grid line to where it crosses the x-axis to find the x-coordinate, −2.5. Follow the grid line to where it crosses the y-axis to find the y-coordinate, −0.5. The ordered pair for the point S is (−2.5,−0.5).
Result
The ordered pair for the point S is (−2.5,−0.5).
Page 89 Exercise 25 Answer
Follow the grid line to where it crosses the x-axis to find the x-coordinate, 1.5. Follow the grid line to where it crosses the y-axis to find the y-coordinate, 2.5. The ordered pair for the point T is (1.5,2.5).
Result
The ordered pair for the point T is (1.5,2.5).
Page 89 Exercise 26 Answer
Follow the grid line to where it crosses the x-axis to find the x-coordinate, −1. Follow the grid line to where it crosses the y-axis to find the y-coordinate, −0.5. The ordered pair for the point U is (−1,−0.5).
Result
The ordered pair for the point U is (−1,−0.5).
Page 89 Exercise 27 Answer
All points in the Quadrant III have both the x-coordinate and the y-coordinate negative. The Fire House is located at (-8,-6), since both -8 and -6 are negative, the Fire House is located in Quadrant III.
Result
The Fire House is located in Quadrant III.
Page 89 Exercise 28 Answer
To find which two places have the same x-coordinates we can look at the x-coordinates of all points.
The x-coordinate of the Fire House is -8, of the School 4, of the Doctor’s Office 3, the Library 10, of the Swimming Pool 12, and of the Club House 12. We can see that both the Swimming Pool and the Club house have the x-coordinate 12.
Result
The Swimming Pool and the Club house have the same x-coordinate.
Page 89 Exercise 29 Answer
First, we need to find the coordinates of the School. Follow the grid lines to directly to the x-axis to find the x-coordinate, 4. Follow the grid lines to directly to the y-axis to find the y-coordinate, 3. The coordinates of the School are (4,3).
To find the point which is the reflection of the point (4,3) across the y-axis simply change the sign of the x-coordinate. Thus, the reflection of the point (4,3) is (−4,3).
Result
The entrance to the new city park is at the point (−4,3).
Page 89 Exercise 30 Answer
If we follow the grid lines the shortest route is either we first go right 3 units from (0,0) to (3,0) and then down 5 units from (3,0) to (3,−5) or we go down 5 units from (0,0) to (0,−5) and then right 3 units from (0,−5) to (3,−5). Either way, the distance is 3 + 5 = 8 units.
Result
Right 3 units and then down 5 units, or down 5 units and then right 3 units, for a total distance of 8 units.
Page 90 Exercise 31 Answer
First, write the coordinates of the ordered pair (-0.7, -0.2) as fractions, -0.7 = –\(\frac{7}{10}\) and -0.2 = –\(\frac{2}{10}\). Follow the grid lines from –\(\frac{7}{10}\) on the x-axis and from –\(\frac{7}{10}\) on the y-axis to where they meet. At that point is the Pond.
Result
At (−0.7,−0.2) is the Pond.
Page 90 Exercise 32 Answer
Follow the grid lines from \(\frac{3}{10}\) on the x-axis and from –\(\frac{2}{10}\) on the y-axis, since –\(\frac{1}{5}\) = –\(\frac{2}{10}\), the where they meet. At that point is the Start of Hiking Trail.
Result
At (\(\frac{3}{10}\), –\(\frac{1}{5}\)) is the Start of Hiking Trail.
Page 90 Exercise 33 Answer
Follow the grid lines directly to the x-axis to find the x-coordinates of the End of Hiking Trail, \(\frac{2}{10}\) which is equal to \(\frac{1}{5}\). Follow the grid lines directly to the y-axis to find the y-coordinates of the End of Hiking Trail, –\(\frac{8}{10}\) which is equal to –\(\frac{4}{5}\).
The coordinates of the End of Hiking Trail can be written in decimal form as (\(\frac{2}{10}\), –\(\frac{8}{10}\))=(0.2,−0.8) or in fraction form as (\(\frac{1}{5}\), –\(\frac{4}{5}\)).
Result
(0.2,−0.8) or (\(\frac{1}{5}\), –\(\frac{4}{5}\))
Page 90 Exercise 34 Answer
To find the coordinates of the information center follow the grid lines directly to the x-axis to find the x-coordinates, –\(\frac{2}{10}\) which is equal to –\(\frac{1}{5}\). Follow the grid lines directly to the y-axis to find the y-coordinates. It is half way between \(\frac{7}{10}\) and \(\frac{8}{10}\), so we add one half of on thenth, \(\frac{1}{2} \times \frac{1}{10}=\frac{1 \times 1}{2 \times 10}=\frac{1}{20}\) to \(\frac{1}{7}\).
\(\frac{7}{10}+\frac{1}{20}=\frac{14}{20}+\frac{1}{20}=\frac{14+1}{20}=\frac{15}{20}=\frac{3}{4}\)The coordinates of the Information Center are (-\(\frac{1}{5}\), \(\frac{3}{4}\)).
Result
(-\(\frac{1}{5}\), \(\frac{3}{4}\))
Page 90 Exercise 35 Answer
First, we must find the coordinates of the pond (or use Exercise 31. on page 90). Follow the grid lines directly to the x-axis to find the x-coordinate, –\(\frac{7}{10}\). Follow the grid lines directly to the y-axis to find the y-coordinate, –\(\frac{1}{5}\). The ordered pair representing the location of the pond is (-\(\frac{7}{10}\), –\(\frac{1}{5}\)).
The reflection of that point across the x-axis is the point which has the same x-coordinate and its y-coordinate has a different sign but the same absolute value. That is the point (-\(\frac{7}{10}\), \(\frac{1}{5}\))
Result
(-\(\frac{7}{10}\), \(\frac{1}{5}\))
Page 90 Exercise 36 Answer
There are four picnic areas. Follow the grid lines directly to the x-axis to find the x-coordinates, and follow the grid lines directly to the y-axis to find the y-coordinates of the picnic areas. Picnic area 1 is located at (-\(\frac{8}{10}\), \(\frac{7}{10}\)), Picnic Area 2 at (\(\frac{6}{10}\), \(\frac{9}{10}\)), Picnic Area 3 at (\(\frac{6}{10}\), \(\frac{3}{10}\)), and Picnic Area 4 at (-\(\frac{8}{10}\), –\(\frac{6}{10}\)).
Points which have {the same x-coordinates} and their y-coordinates are opposites}$, such are the coordinates of Picnic Area 1 and Picnic Area 4, so they are a reflection of each other across the x-axis. None of the other points fit this description.
None of the points have the same y-coordinates and x-coordinates that are opposites, so none of the points are a reflection of each other across the y-axis.
Result
Picnic Area 1 and Picnic Area 4
Page 90 Exercise 37 Answer
To label the points, first write all coordinates as fraction with the same denominator to make it easier.
Since we know \(\frac{3}{4}\) = 0.75, we know that −2.75 can be written as –\(2 \frac{3}{4}\) and −1.75 as –\(1 \frac{3}{4}\). Since \(\frac{1}{4}\) = 0.25, −2.25 can be written as –\(2 \frac{1}{4}\).
\(B(-2.75,-2.25)=\left(-2 \frac{3}{4},-2 \frac{1}{4}\right), \quad D\left(-1 \frac{3}{4}, 2\right)\)To graph the point A follow the grid lines directly from \(\frac{3}{4}\) on the x-axis and from –\(1 \frac{1}{2}\) to where they meet, mark the point and label it A.
To graph the point B follow the grid lines directly from –\(2 \frac{3}{4}\) on the x-axis and from –\(2 \frac{1}{4}\) to where they meet, mark the point and label it B.
Since its x-coordinate is zero, the point C is on the y-axis. Find \(2 \frac{1}{4}\) on the y-axis, mark the point and label it C.
To graph the point D follow the grid lines directly from –\(1 \frac{3}{4}\) on the x-axis and from 2 to where they meet, mark the point and label it D.
Result
To graph each point, follow the grid lines directly from the x-coordinate on the x-axis and from the y-coordinate on the y-axis to where they meet, mark the point and then label it.