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Coordinate plane in four quadrants

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5349306
The graph of a function is shown in the coordinate plane. Read the graph to determine the coordinates of points \(D\), \(E\), and \(F\).
Figure for problem 534930

Hints

- From the origin, read the horizontal coordinate first and the vertical coordinate second. - A point on an axis has one coordinate equal to \(0\).

Solution

1. Point \(D\) is on the x-axis at \(x=-2\), so \(D=(-2, 0)\). 2. Point \(E\) is on the y-axis at \(y=-1\), so \(E=(0, -1)\). 3. Point \(F\) is on the x-axis at \(x=2\), so \(F=(2, 0)\).

Answer

\(D=(-2, 0)\), \(E=(0, -1)\), and \(F=(2, 0)\)
5187306
Point \(A(15, 22)\) lies on a line parallel to the x-axis. Give the coordinates of three different points that also lie on this line.

Hints

- What coordinate is shared by all points on a horizontal line? - Which coordinate changes as you move parallel to the x-axis? - Choose three different values for the x-coordinate.

Solution

1. Every point on a horizontal line has the same y-coordinate. 2. Since \(A\) has y-coordinate \(22\), every point on the line has the form \((x, 22)\). 3. Choose three distinct x-coordinates other than \(15\). For example, \((0, 22)\), \((10, 22)\), and \((30, 22)\) all lie on the line.

Answer

One possible answer is \((0, 22)\), \((10, 22)\), and \((30, 22)\). Any three distinct points of the form \((x, 22)\), different from \(A\), are correct.
5187316
Point \(B(-10, 35)\) lies on line \(h\), which is parallel to the y-axis. Which of the following points also lie on \(h\)? \(P_1(-10, 0)\), \(P_2(10, 35)\), \(P_3(-10, -5)\), \(P_4(0, 35)\), \(P_5(-10, 100)\)

Hints

- Which coordinate stays constant on a vertical line? - Compare the first coordinate of \(B\) with the first coordinate of each listed point. - The y-coordinate can vary on a line parallel to the y-axis.

Solution

1. Every point on a vertical line has the same x-coordinate. 2. Because \(B\) has x-coordinate \(-10\), every point on \(h\) must also have x-coordinate \(-10\). 3. The points with x-coordinate \(-10\) are \(P_1\), \(P_3\), and \(P_5\).

Answer

\(P_1(-10, 0)\), \(P_3(-10, -5)\), and \(P_5(-10, 100)\) lie on line \(h\).
5190766
A world map uses a coordinate system based on longitude and latitude. Longitude is the x-coordinate: east is positive and west is negative. Latitude is the y-coordinate: north is positive and south is negative. The table uses rounded coordinates. | Place | Coordinates \((x, y)\) | | :--- | :--- | | Cairo | \((31, 30)\) | | London | \((0, 51)\) | | Quito | \((-78, 0)\) | | Melbourne | \((145, -38)\) | a) Which place lies on the x-axis, which represents the equator? b) Which place is farthest south? c) Which place is farthest east? d) Which place lies on the y-axis, which represents the prime meridian?

Hints

- Points on the x-axis have y-coordinate \(0\). - Farther south means a smaller y-coordinate. - Farther east means a larger x-coordinate. - Points on the y-axis have x-coordinate \(0\).

Solution

1. A point on the x-axis has y-coordinate \(0\). Quito has coordinates \((-78, 0)\), so Quito lies on the equator. 2. The place farthest south has the least y-coordinate. Since \(-38\) is the least y-coordinate, Melbourne is farthest south. 3. The place farthest east has the greatest x-coordinate. Since \(145\) is the greatest x-coordinate, Melbourne is farthest east. 4. A point on the y-axis has x-coordinate \(0\). London has coordinates \((0, 51)\), so London lies on the prime meridian.

Answer

a) Quito b) Melbourne c) Melbourne d) London
5331436
Find the coordinates of points \(A\) through \(F\) in the graph. For each point, state its quadrant or name the coordinate axis on which it lies.
Figure for problem 533143

Hints

- Read the x-coordinate first and the y-coordinate second. - Quadrants are numbered counterclockwise beginning in the upper-right region. - A point lies on an axis when one coordinate is \(0\).

Solution

1. Reading the graph gives \(A(4, 3)\), \(B(-3, 2)\), \(C(-2, -4)\), \(D(3, -1)\), \(E(0, 2)\), and \(F(-5, 0)\). 2. Point \(A\) is in Quadrant I, \(B\) is in Quadrant II, \(C\) is in Quadrant III, and \(D\) is in Quadrant IV. 3. Point \(E\) lies on the y-axis because its x-coordinate is \(0\). Point \(F\) lies on the x-axis because its y-coordinate is \(0\).

Answer

\(A(4, 3)\), Quadrant I; \(B(-3, 2)\), Quadrant II; \(C(-2, -4)\), Quadrant III; \(D(3, -1)\), Quadrant IV; \(E(0, 2)\), y-axis; \(F(-5, 0)\), x-axis.
5331726
A triangle with vertices \(A\), \(B\), and \(C\) is shown on the coordinate plane. a) Find the coordinates of all three vertices. b) Which point has a negative x-coordinate and a positive y-coordinate?
Figure for problem 533172

Hints

- Read the x-coordinate first and the y-coordinate second. - Coordinates left of the y-axis have negative x-values. - Coordinates above the x-axis have positive y-values.

Solution

1. Reading the graph gives \(A(4, 1)\), \(B(-3, 4)\), and \(C(-2, -5)\). 2. Point \(B\) has a negative x-coordinate and a positive y-coordinate, so it lies in Quadrant II.

Answer

a) \(A(4, 1)\), \(B(-3, 4)\), and \(C(-2, -5)\) b) Point \(B\)
5349126
The vertices of a simple house outline are marked on the coordinate plane. Find the coordinates of points \(P\), \(Q\), \(R\), \(S\), and \(T\).
Figure for problem 534912

Hints

- Read the x-coordinate first. - Then read the y-coordinate. - A point on an axis has one coordinate equal to \(0\).

Solution

1. Reading the graph gives \(P(-2, 0)\), \(Q(2, 0)\), \(R(2, 3)\), \(S(0, 5)\), and \(T(-2, 3)\).

Answer

\(P(-2, 0)\), \(Q(2, 0)\), \(R(2, 3)\), \(S(0, 5)\), and \(T(-2, 3)\)
5187326
Points \(R(7, 12)\) and \(S(7, 40)\) lie on the same line \(g\). a) Is line \(g\) parallel to the x-axis or the y-axis? Explain. b) Point \(T\) lies on \(g\), and its y-coordinate is halfway between the y-coordinates of \(R\) and \(S\). Find the coordinates of \(T\).

Hints

- Compare the two coordinates of \(R\) and \(S\). Which coordinate is the same? - A constant x-coordinate describes what kind of line? - Average the two y-coordinates to find the value halfway between them.

Solution

1. Points \(R\) and \(S\) have the same x-coordinate, \(7\), so the line through them is vertical and parallel to the y-axis. 2. The y-coordinate halfway between \(12\) and \(40\) is \(\frac{12 + 40}{2} = 26\). 3. Every point on \(g\) has x-coordinate \(7\), so \(T = (7, 26)\).

Answer

a) Line \(g\) is parallel to the y-axis because its points have the same x-coordinate. b) \(T = (7, 26)\)
5188906
Line \(g\) passes through \(P(3, 2)\) and \(Q(3, 8)\). Line \(h\) passes through \(R(7, 2)\) and \(S(7, 8)\). One coordinate unit represents \(1\,\text{cm}\). a) Describe the location of all points that are the same distance from \(g\) and \(h\). b) Describe the location of all points that are \(2\,\text{cm}\) from \(g\). Give the equations of the lines.

Hints

- Identify the equations and directions of \(g\) and \(h\). - Find the x-coordinate halfway between the two lines. - Move the given perpendicular distance to both sides of \(g\). - A fixed positive distance from a line produces two parallel lines.

Solution

1. Line \(g\) is \(x = 3\), and line \(h\) is \(x = 7\). Both are vertical and parallel. 2. Points equidistant from two parallel lines lie on the parallel line halfway between them. The halfway x-coordinate is \((3 + 7) \div 2 = 5\), so the locus is \(x = 5\). 3. Points \(2\,\text{cm}\) from \(g\) lie on two vertical lines. Their x-coordinates are \(3 - 2 = 1\) and \(3 + 2 = 5\), so the lines are \(x = 1\) and \(x = 5\).

Answer

a) The points lie on the vertical line \(x = 5\), halfway between \(g\) and \(h\). b) The points lie on the vertical lines \(x = 1\) and \(x = 5\).

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