Understanding X- and Y-Intercepts: Methods & Examples
The general form equation of a line connecting two points (x₁, y₁) and (x₂, y₂) is given as y – y₁={[y₂ - y₁]/[x₂ - x₁]} * (x - x₁). The slope form of a line connecting two points (x₁, y₁) and (x₂, y₂) is equivalent to {y₂ - y₁}/{x₂ - x₁}. Remember that anytime we need to obtain the equation of a line or equation of a line in standard form, we require two things i.e.
A point
A slope
Intercepts Y
The y-intercepts are actually the points where the graph of a function or an equation “touches” or passes through the y-axis in the Cartesian Plane. You may also consider this as a point having x-value of zero.
In order to determine the y-intercepts of an equation, let x = 0, then solve for y.
In a point notation, it is expressed as (0,y)
How to Find the X-Intercepts
Just like the y-intercept, the x-intercepts are basically the points where the graph of a function or an equation “touches” or passes through the x-axis of the Cartesian Plane. Imagine this as a point with y-value of zero.
In order to find the x-intercepts of an equation, let y = 0, then solve for x.
In a point notation, it is expressed as (x, 0).
Finding Intercepts Equation
Let’s first learn how to Find the x and y-intercepts of the general form equation of a line y = –2x + 4.
In order to identify the x-intercepts algebraically, we let y=0 in the equation and then solve for x. Likewise, to find the intercept y algebraically, we let x=0 in the equation and then solve for values of y.
Below is the graph to verify our answers are correct.
How to Find the X and Y-Intercepts of the Quadratic Equation
Let's learn how to determine x and y-intercepts of the quadratic equation. Consider a quadratic equation: y = x² − 2x − 3.
Now, the graph of this quadratic equation will be in the shape of a parabola. We assume it to have a “U” shape in which it would either open up or down.
In order to solve for the x-intercept of this problem, we would require factoring a simple trinomial. Then you set each binomial factor equivalent to zero and solve for value of x.
Below are our solved values for both x and y-intercepts that match along with the graphical solution.
Solved Examples
Example:
Find the intercept of the given function
Determine the intercepts of the equation given as; y=-3x - 4. Then plot the graph with the help of only the intercepts.
Solution:
Set y=0 in order to find out the x-intercept.
y=−3x−4
0=−3x−4
4=−3x
-4/3 = x
= (−4/3) = 0 x intercept
Set y=0 in order to find out the y-intercept.
y=−3x−4
y=−3x(0)−4
y= -4
4=−3x
-4/3 = x
=(0, -4)y intercept
Now, let’s plot both x and y intercept slope intercept form, and draw a line crossing through them as in the figure shown below:
FAQs on How to Find Intercepts From an Equation
1. What are the x-intercept and y-intercept of an equation's graph?
The intercepts are the points where the graph of an equation crosses the coordinate axes.
- The x-intercept is the point where the graph cuts the x-axis. At this point, the value of the y-coordinate is always zero.
- The y-intercept is the point where the graph cuts the y-axis. At this point, the value of the x-coordinate is always zero.
2. What is the standard method to find the intercepts from any given linear equation?
To find the intercepts from an equation, you can follow a simple two-step process, which is a core concept in the CBSE syllabus for the 2025-26 session:
- To find the x-intercept: Set the variable y = 0 in the equation and solve for x. The resulting point will be (x, 0).
- To find the y-intercept: Set the variable x = 0 in the equation and solve for y. The resulting point will be (0, y).
3. How does the slope-intercept form (y = mx + c) make it easier to find the y-intercept?
The slope-intercept form, y = mx + c, is specifically designed to reveal key properties of a line at a glance. In this form:
- 'm' represents the slope of the line.
- 'c' directly represents the y-intercept. This is because if you substitute x = 0 into the equation, you get y = m(0) + c, which simplifies to y = c. This provides a quick way to identify where the line crosses the y-axis without any calculation.
4. What is the difference between the 'x-intercepts' and the 'zeros' of a function?
While closely related, the terms refer to slightly different concepts. The x-intercept is a point on the graph, represented by coordinates (x, 0), where the function's graph crosses the x-axis. The 'zero' or 'root' of a function is the specific x-value at that point. In simple terms, if the x-intercept is (5, 0), then the zero of the function is 5. They are conceptually linked: finding the zeros of a function is the same as finding the x-coordinates of its x-intercepts.
5. What is the practical importance of finding intercepts in real-world graphs, like in physics?
Intercepts often represent significant initial or final conditions in real-world scenarios. For example:
- In a distance-time graph, the y-intercept (at time t=0) represents the initial starting position of an object.
- In a velocity-time graph, the y-intercept represents the initial velocity of the object.
- In a cost function graph (C = ax + b), the y-intercept 'b' represents the fixed cost, which is the cost incurred even when zero items (x=0) are produced.
6. Can a straight line have no x-intercept or no y-intercept?
Yes, this is possible in specific cases. A line will be parallel to an axis and will not intercept it.
- A horizontal line (e.g., y = 4) is parallel to the x-axis and will never cross it, so it has no x-intercept (unless the line is y=0, the x-axis itself).
- A vertical line (e.g., x = 3) is parallel to the y-axis and will never cross it, so it has no y-intercept (unless the line is x=0, the y-axis itself).
7. How does the intercept form of a line (x/a + y/b = 1) relate to its intercepts?
The intercept form, x/a + y/b = 1, is extremely useful because it directly provides the intercepts of the line. In this specific form:
- The value 'a' is the x-coordinate of the x-intercept. The x-intercept point is (a, 0).
- The value 'b' is the y-coordinate of the y-intercept. The y-intercept point is (b, 0).
