The y-intercept of a graph is the point where it crosses the y-axis, which is the vertical axis from the xy-coordinate plane. Below, we will see how to find the y-intercept of any function and why a function can have at most one y-intercept in general. You can also always scroll down to a
video example. advertisement Before we go into detail, consider the graph below. As you can see, it is a linear function (the graph is a line) and it crosses the y-axis at the point (0, 3). This tells you that the y-intercept is 3.
Seeing it on a graph
Since any point along the y-axis has an x-coordinate of 0, the form of any y-intercept is \((0, c)\) for some number \(c\).
Using algebra to find the y-intercept of a function
To find the y-intercept of a function, let \(x = 0\) and solve for \(y\). Consider the following example.
Example
Find the y-intercept of the function: \(y = x^2 + 4x – 1\)
Solution
Let \(x = 0\) and solve for \(y\).
\(\begin{align} y &= 0^2 + 4(0) – 1\\ &= \boxed{-1}\end{align}\)
Thus the y-intercept is –1 and is located at the point \((0, –1)\).
A closer look
Now that we have seen how to find them, there are two interesting questions that can come up:
- Can a function have more than one y intercept?
- Can a function have no y intercept?
In answering these, remember that by definition, a function can only have one output (y-value) for each input (x-value). A function having more than one y-intercept would violate this, since it would mean that there are two outputs for \(x = 0\). Therefore, it is not possible for a function to have more than one y-intercept.
What about no y intercept? Well, consider the graph below. This is a graph of the function: \(y = \dfrac{1}{x}\)
This function never crosses the y-axis because, since you can’t divide by zero, it is undefined at \(x = 0\). In fact, any time a function is undefined at 0, it will have no y-intercept.
Video example
In the video below, I show you three examples of how to find the y-intercept. As you will see, the idea is pretty straight-forward!
Summary
When working with any graph, two useful things to know are the location of any x-intercepts, and the location of the y-intercept, if it exists. With a linear function (a line) these two points are enough to quickly sketch a graph. For more complex functions however, finding intercepts is often part of a deeper analysis.
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Continue your study of graphing
You may find the following articles useful as you continue to study graphs:
- Finding and understanding x-intercepts
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The X-Intercepts
The x-intercepts are points where the graph of a function or an equation crosses or “touches” the x-axis of the Cartesian Plane. You may think of this as a point with y-value of zero.
- To find the x-intercepts of an equation, let y = 0 then solve for x.
- In a point notation, it is written as \left( {x,0} \right).
x-intercept of a Linear Function or a Straight Line
x-intercepts of a Quadratic Function or Parabola
The Y-Intercepts
The y-intercepts are points where the graph of a function or an equation crosses or “touches” the y-axis of the Cartesian Plane. You may think of this as a point with x-value of zero.
- To find the y-intercepts of an equation, let x = 0 then solve for y.
- In a point notation, it is written as \left( {0,y} \right).
y-intercept of a Linear Function or a Straight Line
y-intercept of a Quadratic Function or Parabola
Examples of How to Find the x and y-intercepts of a Line, Parabola, and Circle
Example 1: From the graph, describe the x and y-intercepts using point notation.
The graph crosses the x-axis at x= 1 and x= 3, therefore, we can write the x-intercepts as points (1,0) and (–3, 0).
Similarly, the graph crosses the y-axis at y=3. Its y-intercept can be written as the point (0,3).
Example 2: Find the x and y-intercepts of the line y = - 2x + 4.
To find the x-intercepts algebraically, we let y=0 in the equation and then solve for values of x. In the same manner, to find for y-intercepts algebraically, we let x=0 in the equation and then solve for y.
Here’s the graph to verify that our answers are correct.
Example 3: Find the x and y-intercepts of the quadratic equation y = {x^2} - 2x - 3.
The graph of this quadratic equation is a parabola. We expect it to have a “U” shape where it would either open up or down.
To solve for the x-intercept of this problem, you will factor a simple trinomial. Then you set each binomial factor equal to zero and solve for x.
Our solved values for both x and y-intercepts match with the graphical solution.
Example 4: Find the x and y-intercepts of the quadratic equation y = 3{x^2} + 1.
This is an example where the graph of the equation has a y-intercept but without an x-intercept.
- Let’s find the y-intercept first because it’s extremely easy! Plug in x = 0 then solve for y.
- Now for the x-intercept. Plug in y = 0, and solve for x.
The square root of a negative number is imaginary. This suggests that this equation does not have an x-intercept!
The graph can verify what’s going on. Notice that the graph crossed the y-axis at (0,1), but never did with the x-axis.
Example 5: Find the x and y intercepts of the circle {\left( {x + 4} \right)^2} + {\left( {y + 2} \right)^2} = 8.
This is a good example to illustrate that it is possible for the graph of an equation to have x-intercepts but without y-intercepts.
When solving for y, we arrived at the situation of trying to get the square root of a negative number. The answer is imaginary, thus, no solution. That means the equation doesn’t have any y-intercepts.
The graph verifies that we are right for the values of our x-intercepts, and it has no y-intercepts.