Consider a linear function whose graph is a straight line. But how can we concretely plot the straight line in order to determine its properties?
Gradients and intercepts are the only piece of information we need in order to plot the graph of a straight line and know its properties. Let us explore what they are in this article.
Gradient and intercepts of a line meaning
Gradient of a line meaning
Suppose a man is walking up a hill, and for keeping things simple, the hill is straight in its path with no bumps and curves.
If the man travels 100 meters in the horizontal direction and finds himself 200 meters above the ground, how can we measure the steepness of the hill that is, how inclined the hill is?
A man standing on top of a hill, StudySmarter Originals
To be more rigorous, the sloppiness, in mathematical terms, is known as the Gradient of the hill.
The gradient of a straight line is the measure of how steep it is, or how sloppy it is.
Intercepts of a line meaning
A linear function of the form 
is represented by a straight line that extends infinitely in either direction. Hence, at one point it will intersect either the y-axis or the x-axis or more commonly, both the axes of the coordinate plane.
The coordinates of the point where a straight line intersects either of the axes is known as the intercept of that straight line.
Types of intercepts of a line
And as there are two axes in the coordinate plane, a straight line may have intersections with the x-axis or with the y-axis, or with both of them.
The x-intercepts are the points where a straight line intersects the x-axis.
The y-intercepts are the points where a straight line intersects the y-axis.
Gradient and intercepts calculation and formula
Gradient calculation and formula
The gradient of a line is the ratio of the change in the y-direction to the change in the x-direction.
To measure the steepness of the hill, an intuitive way is to know how the change in the vertical direction is with respect to the change in a horizontal direction, which is essentially what we want to know. And this is exactly how the gradient is defined mathematically.
Translating this definition into an equation, with m denoting the gradient, we get
where 
, 
denote the final and initial y-coordinates, and 
denote the final and initial x-coordinates.
Geometrical interpretation of the gradient
On close inspection, it can be observed that the formula of the gradient which is equal to the ratio between the change in y over the change in x, is nothing but the tangent of the angle formed between the line and the positive x-axis.
Now if we go back to our man on the top of the hill, we can calculate the gradient of the hill with our formula,
Hence, the hill has a gradient of 2.
But how can we find the gradient of a line if we are given nothing but only the equation of the line? According to the definition of the gradient, we recall
.
Hence we need the coordinates of any two points on the line to calculate the gradient.
But there are infinite of them to choose from, and how do we know which ones to choose? Recall that we do know the coordinates of two specific points on the line, the x and y-intercepts!
So we can choose the two points as 
and 
. After substitution we get,
Therefore, the gradient (or slope) of a line given by 
is 
.
Remember that slope and gradient share the same meaning in this context of a straight line.
Intercepts calculation
We consider the graph of a straight line intercepting the axes of the coordinate plane.
The graph of a linear function ax+by+c=0, StudySmarter Originals
We notice that as per the definition, the intercepts are formed on the axes themselves.
The x-intercept will lie on the x-axis and so the y coordinate will be 0 at that point.
- To calculate the x-intercept, set y=0 and solve for x.
Consider a straight line, modeled by the linear equation, where a, b and c are real-valued constants, with a and b not being simultaneously 0.
To find the x-intercepts of this line, we need to substitute y by 0 since the y-coordinate vanishes at that point. Hence, we get
Thus, the x-intercept of the line 
, is 
.
- To calculate the y-intercept, set x=0 and solve for y.
To find the y-intercept of the line, we need to substitute x by 0, which yields
Thus, the y-intercept of the line 
is 
.
Gradient and Intercepts examples
Find the x-intercept and the y-intercept of the line of equation
.
Solution
Comparing the given equation of the line with the general form: 
,
We get 
.
And we know that the y-intercept is given by
.
Hence the y-intercept lies at 
.
Similarly, the x-intercept is given by
.
Hence the x-intercept lies at
.
If we were to plot the line, then we locate the x and y-intercepts and then connect them to get the desired line.
The plot of the straight line 2x+4y-1=0, StudySmarter Originals
Find the gradient of the line of equation 
.
Solution
Comparing the given equation of line to the general form 
, we get 
.
The slope or the gradient of the line can be calculated via,
Thus, the slope of the given line is 3.
The graph of this straight line would be,
The graph of the straight line given by 3x-y+1=0, StudySmarter Originals
where A and B lie at the x and y-intercepts of the line.
Recall that the coordinates of the x-intercept are 
and for the y-intercept 
.
Using this, the x-intercept of our line 
is 
and the y-intercept is 
.
Gradient and Intercepts applications: Slope-Intercept form of a line
Just as a straight line can be generally expressed by the form 
, we can also derive a general form determined by the slope and the intercept of the line.
If we rearrange the given equation to get y on one side of the equation, we have
where we observe that 
is the slope of the line as we found out in the previous section. And let us name 
where d is just another constant renamed in terms of c and b. Recall that 
is the y-intercept itself, which here will be d. Hence, our equation is reduced to
Find the slope and the y-intercept of the line 
.
Solution
To compare the given equation of the line with the slope- intercept form, we need to solve it explicitly for y, we have
Dividing by 2, we get,
Comparing this with the standard form 
, where m is the slope and d is the y -intercept, we get 
.
Hence the slope is 
and the y-intercept is 
.
To find the x-intercept, we set y=0, and we solve for x, and in this case we get,
and thus the x-intercept is 
.
Find the slope and the y-intercept of the line
.
Solution
Comparing the given equation to the general form 
, we get 
.
The slope-intercept form is given by 
, which gives us 
.
Thus the slope is 
and the y-intercept is (0,0).
In order to find the x-intercept, we set y=0 and solve for x. Thus, we get
and hence the x-intercept is (0,0).
Gradient and Intercepts - Key takeaways
- The non-zero coordinates of the point where a straight line intersects the two axes are known as the intercepts of that line.
- For a line given by
, the x-intercept is given by 
and the y-intercept as 
. - The gradient of a line is a measure of how steep it is (or sloppy it is). An alternative term for gradient is slope.
- The gradient of a straight line given by
where 
is the angle the line makes with the positive x-axis. - A straight line can be alternatively expressed in a slope-intercept form where we can deduce the slope and the y-intercept directly.
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