Coordinate geometry Flashcards
Basics
Imagine a big plus sign (+) drawn on a piece of paper.
The horizontal line is called the x-axis (think of it like a road).
The vertical line is called the y-axis (like a tall building).
Where they meet in the middle is called the origin (0,0)—think of it as “home base.”
The Four Quadrants
Quadrant 1 (I): Top-right → Both x and y are positive (like +3, +2)
Quadrant 2 (II): Top-left → x is negative, y is positive (like -3, +2)
Quadrant 3 (III): Bottom-left → Both x and y are negative (like -3, -2)
Quadrant 4 (IV): Bottom-right → x is positive, y is negative (like +3, -2)
Think of it like this:
📍 If you move right, x is positive. If you move left, x is negative.
📍 If you move up, y is positive. If you move down, y is negative.
Plotting Points (Ordered Pairs)
Every point on the grid is written as (x, y)
First number (x) → Tells you how far left or right to go
Second number (y) → Tells you how far up or down to go
Example:
📌 Point (4,2) → Move 4 steps right, then 2 steps up
📌 Point (-4,2) → Move 4 steps left, then 2 steps up
📌 Point (-4,-2) → Move 4 steps left, then 2 steps down
📌 Point (4,-2) → Move 4 steps right, then 2 steps down
Reflections (Flipping Points)
Think of a mirror:
Flip over the x-axis (horizontal) → The y-value changes sign
Flip over the y-axis (vertical) → The x-value changes sign
Flip over the origin (both axes) → Both x and y change signs
Example:
✨ If you have J(4,2):
Mirror over x-axis → (4,-2)
Mirror over y-axis → (-4,2)
Mirror over the origin → (-4,-2)
Finding the distance between two points
We’re finding the distance between two points on a grid (called the xy-plane).
Instead of measuring the distance directly, we pretend it’s the longest side of a right triangle (a hypotenuse).
Imagine you have Point Q (-2, -3) quadrant 3, and Point R (4, 1.5) quadrant 1. You want to know how far apart they are.
We need to measure how far we moved left/right and how far we moved up/down:
Horizontal distance (left to right) → 4 - (-2) = 6
Vertical distance (down to up) → 1.5 - (-3) = 4.5
then, use the pythagorian system:
A² + B² = C²
(where C is the diagonal/hypotenuse).
Square both sides:
6^2 = 36
4.5^2 = 20.25
Add them together:
36+20.25=56.25
Take the square root:
√56.25 = 7.5
Graphing Linear Equations
A linear equation (e.g., y = mx + b) makes a straight line on a graph.
m = slope (rise/run)
b = where the line crosses the y-axis
Intercepts
x-intercept: where the line crosses the x-axis (y = 0)
y-intercept: where the line crosses the y-axis (x = 0)
Think of these like the places where your line “touches” the edges of the graph.
Slope
how steep a line is.
slope= run/rise. = y2 - y1 / x2 - x1
Flat (horizontal) line → Slope = 0 (because it doesn’t rise at all). Equation: y = b
Vertical line → Slope = undefined (because it doesn’t run left or right). Equation: x = a
Graphing Inequalities (Shading)
Now, let’s talk about inequalities (like y > mx + b or y < mx + b).
If it’s y > or y ≥, shade above the line.
If it’s y < or y ≤, shade below the line.
If the inequality has ≥ or ≤, the line is solid.
If it has just > or <, the line is dashed.
Let’s go step by step with an example:
Graph the linear inequality
y < 2x - 3
Step 1: Graph the boundary line
- The inequality is y < 2x - 3
- The related equation is y = 2x - 3
(replace < with = to find the boundary line). - This is a linear equation in slope-intercept form:
- Slope (m) = 2 (rise over run: up 2, right 1)
- Y-intercept ( b ) = -3 (the point (0, -3)
- Plot the y-intercept (0, -3) on the graph.
- Use the slope (rise 2, run 1) to plot another point (e.g. (1, -1) ).
- Draw a dashed line through these points because the inequality does not include the boundary line (no “equal to” part).
Step 2: Shade the correct side
- Pick a test point not on the boundary line (e.g., (0,0) .
- Substitute into the inequality:0 < 2(0) - 3
0 < -3 (False) - Since the test point does not satisfy the inequality, shade the other side of the line.
- If the inequality had been y > 2x - 3, we would shade above the line instead.
Final Graph
- Dashed line for ( y = 2x - 3 )
- Shade below the line (because of “<”).
Graphing Systems of Equations
Graphing a System of Equations
steps:
1. Write in Slope-Intercept Form: Ensure equations are in y=mx+b form.
- Graph Each Line:
Plot the y-intercept (b).
Use the slope (m = rise/run) to plot more points.
Draw a solid line through the points. - Find the Intersection:
The point where both lines cross is the solution.
If they don’t intersect → No Solution.
If they overlap completely → Infinite Solutions.
Graphing Quadratic equations
Let’s break it down step by step for the quadratic equation:
y = x^2 - 2x - 3
Step 1: Find the X-Intercepts (Where the Parabola Crosses the X-Axis)
The x-intercepts are the points where y = 0 (the points where the graph touches the x-axis).
To find them, set y = 0 and solve: x^2 - 2x - 3 = 0
Factor the quadratic equation: (x - 3)(x + 1) = 0
Now, set each factor equal to 0:
x - 3 = 0 → x = 3
x + 1 = 0 → x = -1
So, the x-intercepts are (-1, 0) and (3, 0).
Step 2: Find the Vertex (Lowest or Highest Point of the Parabola)
The vertex is found using the formula for the x-coordinate of the vertex:
x = -b/2a
For y = x^2 - 2x - 3 :
a = 1
b = -2
x =-(-2) / 2(1) = 2/2 = 1
Now, plug x = 1 into the equation to find the y-coordinate:
y = (1)^2 - 2(1) - 3
y = 1 - 2 - 3 = - 4
So, the vertex is at (1, -4).
Step 3: Find the Axis of Symmetry (The Line That Splits the Parabola in Half)
The axis of symmetry is always the vertical line that passes through the x-coordinate of the vertex.
Since we found the vertex at (1, -4) , the axis of symmetry is:
x = 1
This means the left and right sides of the parabola are mirror images around x = 1.
Step 4: Find the Y-Intercept (Where the Parabola Crosses the Y-Axis)
The y-intercept is found by setting x = 0 and solving for y:
y = (0)^2 - 2(0) - 3
y = -3
So, the y-intercept is (0, -3)
Final Answers:
Intercepts: (-1,0) and (3,0)
Vertex: (1, -4)
Axis of Symmetry: x = 1
Y-Intercept: (0, -3)
Graphing circles
The graph of an equation of the form
(𝑥−𝑎)2+(𝑦−𝑏)2=𝑟2
is a circle with its centre at the point with coordinates, a, comma b(a, b)
and with radius r is greater than 0, 𝑟>0.
To graph a circle, follow these steps:
- Use the Standard Equation:(𝑥−𝑎)² +(𝑦−𝑏)² = r²,
where- (a, b)is the center of the circle
- r is the radius
- Identify the Center and Radius:
- If the equation is (x - 2)² + (y + 3)² = 9, then:
- Center = (2, -3)
- Radius = √9 = 3
- If the equation is (x - 2)² + (y + 3)² = 9, then:
- Plot the Center: Mark the point (a, b) on the graph.
- Use the Radius to Find Key Points:
- Move r units up, down, left, and right from the center.
- For (2, -3) with r = 3:
- Up: (2, 0)
- Down: (2, -6)
- Left: (-1, -3)
- Right: (5, -3)
- Sketch the Circle: Connect these points with a smooth, round curve.
Graphing functions
Graphing a function means plotting input-output pairs on the xy-plane to show how the function behaves. Here’s how you do it step by step:
Step 1: Make a Table of Values
Pick some x-values (inputs), plug them into the function, and find the corresponding y-values (outputs).
Step 2: Plot the Points
On the xy-plane, mark the points (-2,2), (-1,-1), (0,-2), (1,-1), and (2,2).
Step 3: Connect the Points Smoothly
Since this is a quadratic function (x²), the points form a parabola (U-shape).
📌 Key Tip:
Linear (y = mx + b) → A straight line
Quadratic (y = ax² + bx + c) → A U-shape (parabola)
Absolute Value (y = |x|) → A V-shape
Square Root (y = √x) → A sideways U
Slope (with more than one point)
M = y2 - y1 / x2 - x1
Intercept
Y= Mx + b
