Graphing Linear Functions: Slope-Intercept Method

1. The Anatomy of a Straight Line: y = mx + c

Any straight line (that isn't vertical) can be described by a simple equation. In your British-based schooling, you would have seen this as:

$y=mx+c$

This is known as the gradient-intercept form. It's incredibly useful because it tells you everything you need to know to draw the line with just a glance.

• c is the y-intercept: This is the point where the line crosses the vertical y-axis. Its coordinate is always $(0,c)$.
• m is the gradient (or slope): This is a measure of the line's steepness and direction.

Let's watch a short video that introduces these concepts clearly. It uses the British convention y = mx + c, which you're familiar with.

A Note on Terminology

As the video showed, the gradient m tells you the "rise over run" – how many units the line goes up for every one unit it goes across to the right.

You should also be aware that in many American resources, the gradient-intercept form is written as y = mx + b. Here, b serves the exact same purpose as c—it represents the y-intercept. We will stick with y = mx + c, but it's important to recognise this alternative notation.

The resource from Math Angel provides a good written summary of these core ideas.

2. The Step-by-Step Graphing Method

Now, let's turn this theory into a practical, repeatable method. To graph a line using its gradient and y-intercept, you only need two points. The y = mx + c form gives us a simple way to find them.

The Method:

1. Plot the Point c: Find the value of c in your equation and place a dot at that value on the y-axis. This is your starting point, $(0,c)$.
2. Use the Gradient m: From your starting point, use the gradient m (rise/run) to find a second point.
   • The numerator (top number) tells you how many steps to move up (if positive) or down (if negative).
   • The denominator (bottom number) tells you how many steps to move to the right.
   • If m is an integer, like 3, you can write it as $\frac{3}{1}$. This means "move 3 up and 1 to the right".
   • If m is negative, like $-\frac{2}{3}$, you can read this as "move 2 down and 3 to the right".
3. Draw the Line: Connect the two points with a straight line and extend it in both directions.

The image below shows this process for the equation $y=0.5x+4$. The gradient, 0.5, is the same as $\frac{1}{2}$, meaning "1 up for every 2 across". (Note: The image uses 0.5 up for 1 across, which is equivalent).

This next video provides an excellent, detailed walkthrough of this method with multiple examples, including how to handle negative gradients.

Here is another visual example, this time with a negative gradient.

3. When the Equation Isn't in the Right Form

Often, linear equations aren't given to you neatly in the y = mx + c form. You might see something like 4x + 2y = 8.

The strategy here is simple: rearrange the equation to isolate y on one side. This puts it into the familiar gradient-intercept form, allowing you to easily identify m and c. This process will draw on the skills you developed in Module 1 for solving linear equations.

Let's return to the kevinmathscience video, which covers this exact scenario.

For more practice and a comprehensive set of examples on rearranging equations, the following article is an excellent resource. It uses the American y = mx + b form, but the algebraic steps for isolating y are universal.

Conclusion

Today we've established a fundamental and highly practical skill: graphing linear functions. By understanding the y = mx + c form, you can quickly deconstruct any linear equation to find its core visual properties.

Key Takeaways:

• The equation of a straight line can be written as y = mx + c.
• c is the y-intercept, your starting point on the y-axis $(0,c)$.
• m is the gradient (slope), which tells you the steepness and direction ("rise over run").
• The process for graphing is: Plot c, use m to find a second point, and draw the line.
• If an equation is not in the correct form, use algebra to isolate y first.

Preview of the Next Lesson:
Now that we know how to graph a basic function like a line from scratch, we'll move on to a more general concept. In the next lesson, "Perform transformations of functions, including vertical and horizontal translations (shifts)," we will learn rules that allow us to take the graph of any function and move it up, down, left, or right on the coordinate plane.