Linear graphs may be one of the first few topics you learn in secondary school math, but that doesn’t mean it’s always easy! To beginners at algebra and graphs, this topic can prove to be a little tricky, especially when doing it for the first few times.
If you’re still at the stage of trying to grapple with linear graphs, here’s a quick refresher for you!
What are linear graphs?
Linear graphs are graphs that are a straight line. So, any graph you draw with a ruler across the graph paper will give you a linear graph. For every value of x, there will be a real value for y.
Some features you can use to describe a linear graph are:
- Gradient: how steep the slope of the graph is
- x-intercept: x coordinate of the point where the line crosses the x-axis
- y-intercept: y coordinate of the point where the line crosses the y-axis
4 ways to write linear graph equations
Generally, any linear equation will form a linear graph. However, to help us find important information at a glance, it is useful to know how to rearrange the linear equation into different forms.
-
Gradient-intercept form
This is a handy form of the linear equation when you need to find the gradient or y-intercept at a glance. It is also pretty neat and you can plot out the graph easily with the information already visible in the equation.
The one thing that is less obvious from this form is the x-intercept. To find the x-intercept, all you need to do is substitute y=0 to find out the value at which the line crosses the x-axis.
-
Point-gradient
When you only have one point and the gradient of the graph, you can easily form this equation. What about when you have two points on the graph but no gradient? You can calculate the gradient and substitute 1 point and the gradient into this form too.
With the available information, drawing out the graph is no problem. When you need to solve the problem without drawing the graph, you can manipulate the equation into y=mx+c form to find the y-intercept.
Like with the gradient-intercept form, you’ll have to substitute y=0 to find the x-intercept.
-
Double-intercept form
In some cases, you’ll only have the x-intercept and y-intercept available to you. This is enough to give you the double-intercept form.
You may also be asked to convert a linear equation into the double-intercept form. To do this, you’ll need to make both sides of the equation equal to 1.
-
General form
ax + by + c = 0
a, b, c, are integers.
This form looks neat, but it is generally not very helpful as it does not tell you anything about the gradient, y-intercept, or x-intercept.
However, some questions may ask you to answer in general form, or provide the general form and ask you to convert it into another form. Hence, it is important to be aware of what the general form is and how to manipulate it to get the other forms of linear equations.
Finding the gradient
You’ll notice how both the gradient-intercept and point-gradient form has the gradient m. As the gradient is a crucial piece of information in any linear graph, you need to be very confident about knowing how to calculate it.
It also helps to draw out a triangle to visualise the rise and run better, as shown above in the graph!
Other ways of finding the gradient
Sometimes, you can’t calculate the gradient from the formula above. But the question may give you other clues, from which you can figure out the gradient.
For example,
- Line A and Line B are parallel: this means they have the same gradient.
- Line A and Line B are perpendicular: this means that their gradients are negative reciprocals of each other (e.g. 2 and -0.5)
- Points P, Q, and R are collinear: this means that the gradient between any of the two points will be the same (as all the points lie on the same straight line)
Conclusion
Linear graph is one of the foundational topics every math student needs to master in order to move on to more difficult topics in secondary school math. That is why if you are having trouble with it, we strongly recommend you to clarify your doubts ASAP!
Our math tutors are highly experienced in teaching secondary school math tuition, and can help you understand complex topics more easily. Want to get more guidance on math topics like this and more? Join us for our small group classes to dispel your fear and doubts surrounding math! Psst – we also have IP Math Tuition classes for those of you in IP schools!