Introduction to Coordinate Geometry
Coordinate Geometry: Unlocking the Map 🗺️
Introduction
1. Introduction
Alright squad, let's talk coordinate geometry. Sounds intense, right? But honestly, it's just the math behind every map you've used in a video game or seen on Snapchat. It's all about plotting points and drawing lines. Think of it as giving instructions to a character: 'go right 3, up 5'. Once you get the hang of the main formula, , you'll be navigating these graphs like a pro. Let's level up! 🚀
2. The Cartesian Coordinate System
First up, the basics. The graph we use is called a Cartesian grid. It's basically a map with two main roads: the horizontal x-axis and the vertical y-axis. The point where they cross is called the origin, which is at . Any point on this map can be found using coordinates, written as . The first number tells you how far to go right (positive) or left (negative), and the second number tells you how far to go up (positive) or down (negative). Easy peasy.
The order always matters: and are different points. A good way to remember it is 'along the corridor, then up or down the stairs'.
Before you read a point off a grid, check the scale on each axis: one square is not always one unit.
An equation like is a rule that links and . For any you choose, it tells you the that goes with it. Find a few of these pairs, plot them, and they line up in a perfectly straight line.
An equation like is a rule that links and . For any you choose, it tells you the that goes with it. Find a few of these pairs, plot them, and they line up in a perfectly straight line.
Worked example
Worked Example: Plotting Points on the Cartesian Plane
Worked Example: Plotting Points
Plot the points and on a grid.
- 1For point : The x-coordinate is 2, so we start at the origin and move 2 units to the right.
- 2The y-coordinate is 5, so from our new x-position, we move 5 units up. Place a dot and label it 'A'.
- 3For point : The x-coordinate is -3, so we start at the origin and move 3 units to the left.
- 4The y-coordinate is 1, so from there, we move 1 unit up. Place a dot and label it 'B'. You've just plotted the points!
Answer
Worked example
Plotting a Linear Graph from a Table of Values
From Equation to Epic Line Graph 📈
Complete the table of values for the equation and then plot the graph for values from to .
x | -2 | -1 | 0 | 1 | 2 | 3
y | | -3 | | 1 | | 5
x | -2 | -1 | 0 | 1 | 2 | 3
y | | -3 | | 1 | | 5
- 1First, we need to fill in the blanks in our table. We do this by substituting the values into our equation, , one by one. Let's find the value for :
- 2Now we do the same for the other missing values. For :
And for : - 3Awesome! Our table is complete. Now, let's list our coordinates as pairs. Think of them as a squad of points ready to be plotted.
- 4Time to draw the axes. Look at your coordinate values. The values go from -2 to 3, and the values go from -5 to 5. Make sure your axes are long enough to fit all these points comfortably: label the x-axis from to and the y-axis from to , with an even scale on each.
- 5Let's plot these points. For each pair, remember: 'along the corridor, then up/down the stairs'. For , we start at the origin, go 2 units left, and then 5 units down. Mark the spot with a small dot or cross. Repeat for all six points.
- 6The grand finale! Grab a ruler and connect all the points. If you've done it right, they should all line up perfectly. Draw a single, straight line that goes through all of them, extend it a little past your first and last points, and label it .
Answer
3. Understanding the Gradient of a Line
The gradient of a line is just a fancy word for its steepness. Think about a hill. A super steep hill is hard to climb, while a gentle slope is easy. In math, we give this steepness a number, which we call . A positive gradient () goes uphill from left to right. A negative gradient () goes downhill. A flat, horizontal line has a gradient of 0 (no effort!). The classic way to remember it is 'rise over run': how much the line goes up or down for every step it takes to the right.
At Core you always find the gradient from a grid: the line is drawn for you. Pick two points where the line passes exactly through grid corners, as far apart as you can. Draw a right-angled triangle between them. The 'rise' is how many units you go up or down, and the 'run' is how many units you go across to the right.
Then . If you write the two grid points as and , that division is : just subtract in the same order top and bottom, and check the scale on each axis before you count. Beyond Core: working out a gradient from two coordinates with no graph at all is Extended content (E3.3.2).
Worked example
Worked Example: Gradient of a Line Drawn on a Grid
Worked Example: Calculating Gradient from a Grid
A straight line is drawn on a grid where one square is one unit. It passes exactly through the grid corners and . Find its gradient.
- 1Draw a right-angled triangle between the two grid corners. First, count the 'rise' (the change in y): from a y-value of 2 up to 8.Rise =
- 2Next, find the 'run' (the change in x). We went from an x-value of 1 to 4.Run =
- 3Now, we just divide the rise by the run to get our gradient, .
Answer
4. The Equation of a Straight Line: y = mx + c
Okay, this is the cheat code for straight lines: . Seriously, almost everything in this chapter comes back to this equation. We already know is the gradient. The new character here is , which stands for the y-intercept. This is simply the point where the line crosses the vertical y-axis. It's the line's 'starting point' on that axis. If you know a line's gradient () and its y-intercept (), you know the equation for the entire line. It's the line's unique ID. If the equation is written the other way round, such as , rewrite it as before you read off and .
Two lines get their own rule. A horizontal line has gradient 0, so its equation is : the line through and is . A vertical line cannot be written as at all; every point on it has the same -value, so its equation is . The vertical line through is .
Two lines get their own rule. A horizontal line has gradient 0, so its equation is : the line through and is . A vertical line cannot be written as at all; every point on it has the same -value, so its equation is . The vertical line through is .
Worked example
Worked Example: Identifying Gradient and y-intercept
Worked Example: Interpreting y = mx + c
A line has the equation . What is its gradient and y-intercept?
- 1We just need to compare the given equation to the master code, .
- 2The number multiplying the is our gradient, . In this case, it's -3.
- 3The number added or subtracted at the end is our y-intercept, . Here, it's +5.
Answer
5. Finding the Equation of a Line from Its Graph
This is the commonest question in the whole chapter, and it's totally doable. When the line is drawn on a grid for you:
1. Read where the line cuts the y-axis.
2. Find with a rise-over-run triangle between two grid corners on the line.
3. Write with your two numbers in, fully simplified.
4. Test a third point from the line in your equation. If it works, you're almost certainly right. 🎯
Sometimes you are given the gradient and one point that is not on the y-axis. Then is unknown: write with the gradient in place, substitute the point's and (the point is on the line, so it must fit), and solve for .
1. Read where the line cuts the y-axis.
2. Find with a rise-over-run triangle between two grid corners on the line.
3. Write with your two numbers in, fully simplified.
4. Test a third point from the line in your equation. If it works, you're almost certainly right. 🎯
Sometimes you are given the gradient and one point that is not on the y-axis. Then is unknown: write with the gradient in place, substitute the point's and (the point is on the line, so it must fit), and solve for .
Worked example
Worked Example: Determining the Equation of a Line from a Grid
Let's Cook Up This Equation 🍳
A straight line is drawn on a grid. It passes through the y-axis at the point (0, -2) and also passes through the point (3, 4). Find the equation of the line in the form .
- 1First, let's find the y-intercept, ''. This is the easiest mark you'll ever get! The y-intercept is the point where the line crosses the y-axis, which is always where . The problem tells us it passes through (0, -2).
- 2Next up, we need the gradient, ''. We'll use the 'rise over run' method between the two points we know: (0, -2) and (3, 4). The 'rise' is the change in the y-coordinates.
- 3Now for the 'run', which is the change in the x-coordinates. Remember to go in the same direction as you did for the rise.
- 4We can now calculate the gradient '' by dividing the rise by the run.
- 5Finally, we assemble our equation. We have our gradient and our y-intercept . We just need to substitute these values into the standard equation .
Answer
6. Properties of Parallel Lines
What does it mean for two lines to be parallel? Think of train tracks or two people scrolling through TikTok side-by-side. They're going in the exact same direction and will never, ever cross. In coordinate geometry, this means they have the exact same gradient. If line 1 has a gradient of and line 2 has a gradient of , then the lines are parallel when . So a line parallel to has gradient 3 and an equation like . The 'something else' is the y-intercept (), which just slides the line up or down without changing its steepness.
Parallel lines have the same and a different . If and both match, the two equations describe the same line, which meets itself everywhere, so it is not a parallel line. Watch out for a gradient of in a list of options: it has the same steepness but slopes the other way, so it crosses the original line.
Worked example
Worked Example: Identifying Parallel Lines Using Gradients
Worked Example: Identifying Parallel Lines
Are the lines and parallel?
- 1First, let's find the gradient of the first line by comparing it to .For , the gradient .
- 2Now, let's find the gradient of the second line.For , the gradient .
- 3We compare the gradients. Since , the gradients are the same.Yes, the lines are parallel.
Answer
Yes, the lines are parallel.
7. Finding the Equation of a Parallel Line Through a Point
This is where it all comes together. You'll be given the equation of a line and a single point that is not on that line. Your mission: find the equation of a new line that is parallel to the first one and goes through your given point. It's like finding a new road that runs parallel to a highway, but goes directly through your town.
1. Steal the Gradient: Look at the original line's equation and find its gradient (). Since your new line is parallel, it will have the exact same .
2. Find the new Y-intercept (): Use the you just found and the coordinates of the new point you were given. Substitute them into and solve for your new .
3. Write the Equation: You've got your and your new . Put them together for the final answer.
1. Steal the Gradient: Look at the original line's equation and find its gradient (). Since your new line is parallel, it will have the exact same .
2. Find the new Y-intercept (): Use the you just found and the coordinates of the new point you were given. Substitute them into and solve for your new .
3. Write the Equation: You've got your and your new . Put them together for the final answer.
Worked example
Worked Example: Finding the Equation of a Parallel Line
Copying the Gradient, Finding the 'c' 🕵️♂️
Find the equation of the line that is parallel to the line and passes through the point .
- 1First, we need the gradient of the line we're given. In the equation , the gradient is the '' value. This is the steepness we need to copy.Original line: .
The gradient () is the number multiplying the .
So, . - 2The golden rule is that parallel lines have equal gradients. So, our new line will have the exact same gradient as the original one.Gradient of our new parallel line is also .
- 3Now we know our new equation looks like . To find the y-intercept, '', we substitute the coordinates of the point we're given, , into this equation. We're basically telling the line where it needs to be.
- 4Time for some quick algebra to solve for . This will pinpoint the exact location of our new line on the graph.
- 5We've got our gradient () and our y-intercept (). Let's assemble them into the final equation. Mission complete!
Answer
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