Working in the Cartesian Plane

Ever wondered why some lines go straight across or straight up on a graph? Horizontal lines run parallel to the x-axis and have equations like y = -1, which means every single point on that line has the same y-coordinate. It's like drawing a flat line across the page.

Vertical lines work the opposite way - they're parallel to the y-axis with equations like x = 3. Every point on a vertical line shares the same x-coordinate, creating a line that goes straight up and down.

The y = x line is special because it creates a perfect 45° angle when your graph scales are equal. At any point on this line, both coordinates match - like (2,2) or $-6,-6$. Think of it as the line where x and y are best friends who always have the same value.

Quick Tip: Remember that the scale on your axes matters! If your x and y scales are different, your y = x line won't look like a perfect 45° angle.

Lines of the Form y = kx

The equation y = kx is where things get interesting - it shows how one variable depends on another. When you see y = 2x, you're basically saying "whatever x is, multiply it by 2 to get y." So if x = 3, then y = 6.

The value of k determines the steepness of your line. A bigger k value creates a steeper line that shoots up quickly, while a smaller k value gives you a gentler slope that hugs closer to the x-axis.

Direct proportion happens when two variables increase at exactly the same rate. Your graph must be a straight line passing through the origin for this to work - if it's curved or wobbly, the variables aren't proportional.

Real-World Connection: Think about hourly wages - if you earn £10 per hour, your total pay $y$ equals £10 times hours worked $x$. That's y = 10x in action!

Lines in the Form y = x + a and y = mx + c

Lines like y = x + 6 and y = x - 4 are just the basic y = x line that's been shifted up or down the graph. They're all parallel because they have the same gradient - they just start from different positions.

The "a" value shows translation - how far up or down the line has moved. If it's y = x + 5, you've moved the y = x line up by 5 places. If it's y = x - 2, you've dropped it down by 2 places.

For y = mx + c equations, create a table with x-values, multiply each by m, then add c. Each pair gives you coordinates to plot. More points mean more accuracy, so don't be lazy - plot at least three points and check they form a straight line.

Exam Tip: Always join your plotted points with a straight line using a ruler. Wobbly freehand lines will cost you marks, even if your calculations are spot-on!

Lines with Negative Gradients

When your line equation has a negative x value (like y = -x or y = -2x), you get a negative gradient that slopes downwards from left to right. Instead of climbing up the graph, these lines take a downward path.

Negative gradient lines always follow the same pattern - they start high on the left and finish low on the right. The steeper the negative number, the more dramatic the downward slope becomes.

Memory Trick: Think "negative = downhill" - negative gradients always slide downwards as you move from left to right across the graph.