Solved Exercises on Graphing Points in Four Quadrants in Grade 7

Master coordinate plane fundamentals: ordered pairs, quadrant identification, and point plotting through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Plotting Points in Different Quadrants
Exercise 1
Plot the following points on the coordinate plane and identify which quadrant each point lies in: A(3, 4), B(-2, 5), C(-4, -3), D(5, -2).
Definition:

Ordered pair: A pair of numbers (x, y) that represents a point on the coordinate plane. The coordinate plane is divided into four quadrants by the x-axis and y-axis.

Point plotting method:
  1. Start at the origin (0, 0)
  2. Move horizontally according to the x-coordinate (right for positive, left for negative)
  3. Move vertically according to the y-coordinate (up for positive, down for negative)
  4. Mark the point at the final location
Point Coordinates
A(3,4), B(-2,5), C(-4,-3), D(5,-2)
Quadrant Signs
I(+,+), II(-,+), III(-,-), IV(+,-)
Quadrant Location
A(I), B(II), C(III), D(IV)
Step 1: Analyze point A(3, 4)

x = 3 (positive), y = 4 (positive) → Both coordinates positive → Quadrant I

Step 2: Analyze point B(-2, 5)

x = -2 (negative), y = 5 (positive) → x negative, y positive → Quadrant II

Step 3: Analyze point C(-4, -3)

x = -4 (negative), y = -3 (negative) → Both coordinates negative → Quadrant III

Step 4: Analyze point D(5, -2)

x = 5 (positive), y = -2 (negative) → x positive, y negative → Quadrant IV

A(3,4) in Quadrant I, B(-2,5) in Quadrant II, C(-4,-3) in Quadrant III, D(5,-2) in Quadrant IV
Final answer:

A(3,4) is in Quadrant I, B(-2,5) is in Quadrant II, C(-4,-3) is in Quadrant III, D(5,-2) is in Quadrant IV

Applied rules:

• Quadrant I: (+, +) - both coordinates positive

• Quadrant II: (-, +) - x negative, y positive

• Quadrant III: (-, -) - both coordinates negative

• Quadrant IV: (+, -) - x positive, y negative

2 Finding Coordinates and Distances
Exercise 2
Point P is located 5 units to the left of the y-axis and 3 units below the x-axis. Find the coordinates of point P and calculate its distance from the origin.
Definition:

Distance from origin: The distance from any point (x, y) to the origin (0, 0) is calculated using the distance formula: d = √(x² + y²)

Position Description
5 units left, 3 units down
Coordinate Conversion
x = -5, y = -3
Distance Formula
d = √(x² + y²)
Step 1: Interpret the position description

"5 units to the left of the y-axis" means x-coordinate is -5

"3 units below the x-axis" means y-coordinate is -3

Step 2: Determine the coordinates

Point P has coordinates (-5, -3)

Step 3: Apply the distance formula

d = √(x² + y²) = √((-5)² + (-3)²)

d = √(25 + 9) = √34 ≈ 5.83 units

Step 4: Verify quadrant location

Both coordinates are negative, so P(-5, -3) is in Quadrant III

P(-5, -3), Distance from origin = √34 ≈ 5.83 units
Final answer:

The coordinates of point P are (-5, -3) and its distance from the origin is √34 ≈ 5.83 units

Applied rules:

• Direction interpretation: Left = negative x, Right = positive x, Up = positive y, Down = negative y

• Distance formula: d = √(x² + y²)

• Quadrant identification: Based on signs of coordinates

3 Identifying Geometric Shapes
Exercise 3
Plot the points A(2, 3), B(-2, 3), C(-2, -3), and D(2, -3) on the coordinate plane. Connect the points in order to form a shape. Identify the shape and calculate its area.
Definition:

Rectangle: A quadrilateral with four right angles and opposite sides equal and parallel. Area of rectangle = length × width.

Vertices
A(2,3), B(-2,3), C(-2,-3), D(2,-3)
Side Lengths
Length = 4, Width = 6
Area Formula
Area = l × w
Step 1: Plot all four points

A(2, 3) in Quadrant I, B(-2, 3) in Quadrant II, C(-2, -3) in Quadrant III, D(2, -3) in Quadrant IV

Step 2: Connect points in order (A→B→C→D→A)

Connecting the points forms a closed shape with four sides

Step 3: Calculate side lengths

Horizontal sides: |2 - (-2)| = 4 units

Vertical sides: |3 - (-3)| = 6 units

Step 4: Verify right angles

All angles are 90°, confirming it's a rectangle

Rectangle with area = 24 square units
Final answer:

The shape formed is a rectangle with an area of 24 square units

Applied rules:

• Rectangle properties: Opposite sides equal, all angles 90°

• Area calculation: Area = length × width

• Distance between points: Use absolute value of coordinate differences

Rules and methods, laws,...
Distance = √(x² + y²)
Distance from Origin
Quadrant I
(+, +)
Positive x, Positive y
Quadrant II
(-, +)
Negative x, Positive y
Quadrant III
(-, -)
Negative x, Negative y
Quadrant IV
(+, -)
Positive x, Negative y
Key definitions:

Coordinate plane: A two-dimensional surface formed by two perpendicular number lines (x-axis and y-axis)

Origin: The point (0, 0) where the x-axis and y-axis intersect

Ordered pair: A pair of numbers (x, y) that represents a point's location on the coordinate plane

Quadrant: One of the four regions created by the intersection of the x-axis and y-axis

Complete methodology:
  1. Identify coordinates: Determine the x and y values from the ordered pair
  2. Locate starting point: Begin at the origin (0, 0)
  3. Move horizontally: Move left for negative x, right for positive x
  4. Move vertically: Move up for positive y, down for negative y
  5. Mark the point: Place the point at the final location
Tip 1: Remember "Run first, then Jump" - x-coordinate (horizontal) comes first, then y-coordinate (vertical).
Tip 2: To identify quadrants quickly, just look at the signs of the coordinates.
Tip 3: Points on axes are not in any quadrant - points on x-axis have y=0, points on y-axis have x=0.
Common errors: Mixing up x and y coordinates, going in wrong direction based on sign, misreading negative numbers.
Exam preparation: Practice plotting many points, memorize quadrant signs, understand distance calculations, recognize geometric shapes on coordinate plane.
Solution: Exercises 4 to 5
4 Reflection Across Axes
Exercise 4
Point M(4, -2) is reflected across the y-axis to form point M'. Then M' is reflected across the x-axis to form point M''. Find the coordinates of M' and M''.
Definition:

Reflection across y-axis: Changes the sign of the x-coordinate: (x, y) → (-x, y). Reflection across x-axis: Changes the sign of the y-coordinate: (x, y) → (x, -y).

Original Point
M(4, -2)
First Reflection
Across y-axis
Second Reflection
Across x-axis
Step 1: Apply reflection across y-axis

Rule: (x, y) → (-x, y)

M(4, -2) → M'(-4, -2)

Step 2: Apply reflection across x-axis to M'

Rule: (x, y) → (x, -y)

M'(-4, -2) → M''(-4, -(-2)) = M''(-4, 2)

Step 3: Verify quadrant locations

M(4, -2) is in Quadrant IV

M'(-4, -2) is in Quadrant III

M''(-4, 2) is in Quadrant II

Step 4: Confirm the transformations

Original point was in Quad IV, after two reflections it ended up in Quad II

M'(-4, -2), M''(-4, 2)
Final answer:

After the reflections, M' has coordinates (-4, -2) and M'' has coordinates (-4, 2)

Applied rules:

• Reflection across y-axis: (x, y) → (-x, y)

• Reflection across x-axis: (x, y) → (x, -y)

• Sequential transformations: Apply transformations in order

5 Distance Between Two Points
Exercise 5
Find the distance between points P(-3, 4) and Q(2, -1) on the coordinate plane. Show your work using the distance formula.
Definition:

Distance formula: The distance between two points (x₁, y₁) and (x₂, y₂) is: d = √[(x₂-x₁)² + (y₂-y₁)²]

Points
P(-3, 4), Q(2, -1)
Distance Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]
Calculated Distance
d = √50 = 5√2
Step 1: Identify coordinates

P(-3, 4): x₁ = -3, y₁ = 4

Q(2, -1): x₂ = 2, y₂ = -1

Step 2: Apply distance formula

d = √[(x₂-x₁)² + (y₂-y₁)²]

d = √[(2-(-3))² + (-1-4)²]

Step 3: Simplify inside brackets

d = √[(2+3)² + (-5)²]

d = √[5² + (-5)²]

d = √[25 + 25]

Step 4: Complete the calculation

d = √50 = √(25×2) = 5√2 ≈ 7.07 units

Distance = 5√2 ≈ 7.07 units
Final answer:

The distance between points P(-3, 4) and Q(2, -1) is 5√2 ≈ 7.07 units

Applied rules:

• Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

• Subtraction with negatives: Subtracting a negative is addition

• Square roots: Simplify when possible

Key Concepts, Laws, Methods, and Definitions
d = √[(x₂-x₁)² + (y₂-y₁)²]
Distance Formula
Key definitions:

Coordinate plane: A two-dimensional surface defined by a horizontal x-axis and vertical y-axis

Ordered pair: A set of two numbers (x, y) representing a point's location

Quadrant: One of four regions created by the intersection of axes

Origin: The reference point (0, 0) where axes intersect

Reflection: A transformation that flips a point across an axis

Complete methodology:
  1. Reading coordinates: Identify x (horizontal) and y (vertical) values
  2. Plotting points: Move horizontally first, then vertically
  3. Identifying quadrants: Based on signs of coordinates
  4. Calculating distances: Use appropriate formulas
Tip 1: Always plot points carefully, paying attention to positive/negative signs.
Tip 2: Remember that distance is always positive, regardless of direction.
Tip 3: For reflections, only one coordinate changes sign depending on which axis is used.
Tip 4: When calculating distances, subtract the first coordinate from the second consistently.
Common errors: Reversing x and y coordinates, going in wrong direction based on sign, calculation errors with negative numbers, misapplying formulas.
Exam preparation: Practice plotting points in all quadrants, memorize quadrant signs, master distance calculations, understand transformations, recognize geometric shapes on coordinate plane.
Coordinate Rules:

• Quadrant I: (+, +) - x > 0, y > 0

• Quadrant II: (-, +) - x < 0, y > 0

• Quadrant III: (-, -) - x < 0, y < 0

• Quadrant IV: (+, -) - x > 0, y < 0

• Reflection across y-axis: (x, y) → (-x, y)

• Reflection across x-axis: (x, y) → (x, -y)

• Distance from origin: d = √(x² + y²)

• Distance between points: d = √[(x₂-x₁)² + (y₂-y₁)²]

Questions & Answers

Question: I keep getting confused about which number in the ordered pair is x and which is y. How can I remember this?

Answer: Here are some helpful memory tricks:

  • Alphabetical order: In the alphabet, x comes before y, just like in ordered pairs (x, y)
  • "X goes first, Y follows": X is the first coordinate, Y is the second
  • "Run first, then Jump": Move horizontally (run) according to x, then vertically (jump) according to y

Think of it as giving directions: "Go this far along (x), then go this far up/down (y)".

Always remember: (x-coordinate, y-coordinate) or (horizontal, vertical)

Question: What happens if one of the coordinates is zero? Where does the point lie?

Answer: Points with a zero coordinate lie on an axis, not in any quadrant:

  • If x = 0 (like (0, 3) or (0, -2)), the point lies on the y-axis
  • If y = 0 (like (4, 0) or (-3, 0)), the point lies on the x-axis
  • If both x = 0 and y = 0 (0, 0), the point is at the origin

Axes are the boundaries between quadrants, so points on axes belong to no specific quadrant.

Example: Point (0, 5) is on the positive y-axis, Point (-4, 0) is on the negative x-axis.

Question: When using the distance formula, does it matter which point I call (x₁, y₁) and which I call (x₂, y₂)?

Answer: No, it doesn't matter which point you designate as (x₁, y₁) or (x₂, y₂) because of squaring in the formula.

Example: Distance between A(1, 2) and B(4, 6)

  • If A is (x₁, y₁) and B is (x₂, y₂): d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5
  • If B is (x₁, y₁) and A is (x₂, y₂): d = √[(1-4)² + (2-6)²] = √[9 + 16] = √25 = 5

Since we square the differences, the order doesn't affect the result: (a-b)² = (b-a)².

However, be consistent within each calculation to avoid sign errors.