Y intercept: Definition, Formula, and Examples - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day

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In defining what is y intercept, we need to take note of the graph of a function. The y-intercept of any given function is the point where the graph touches the y-axis. Thus, the y-intercept of a graph is the point $(0, b)$ where $b$ is the value in the y-axis where the graph crosses.

It is important to solve for the y-intercept of a function because it helps in graphing lines since we already know at what point will the graph cut the y-axis. Moreover, y-intercepts are helpful in other applications of problems involving linear equations.

There are two types of intercepts in a function — we have the x-intercept and y-intercept. Intercepts, in general, are the points where the graph of the function crosses the x-axis or the y-axis. But in this article, we will focus on solving for the y-intercept of a given graph, a given equation, and given any two points in the graph.

How To Find Y-intercept on a Graph

The y-intercept is located at the point in the graph that intersects the y-axis. Here are some examples of locating a y-intercept on a graph.

Example 1

Notice that the graph in Figure 1, the graph of $y = 4 x - 6$ , is a linear function that touches the y-axis at the point $(0, - 6)$ . Thus, the y-intercept of the linear graph $y = 4 x - 6$ is located at the point $(0, - 6)$ .

The graph of the equation $y = 8 - x^{2}$ is a parabola. It intersects the y-axis exactly at the point $(0, 8)$ . Hence, the y-intercept of the parabola $y = 8 - x^{2}$ is at the point $(0, 8)$ .

In general, the y-intercept of a quadratic function is the vertex of the parabola.

The graph of exponential function $y = 1 - e^{x}$ in Figure 3 crosses the y-axis only at the origin and no other point. Therefore, the y-intercept of the given graph is at $(0, 0)$ .

Since we already know how to find y-intercept on a graph, the question now is, “Is it possible for a graph to have no y-intercept?”

Are There Graphs With No Y-intercept?

Yes, it is possible for a graph to have no y-intercept — this means that the graph does not touch the y-axis.

Note that a function satisfies a vertical line test. That is, if we are to draw infinite vertical lines in the graph, each line should touch the graph at most once. Since the y-axis is a vertical line, then the graph touches the y-axis either once or not at all. Moreover, we could note from this that it is not possible for a graph of a function to have more than one y-intercept.

Let’s look at the example of graphs that do not have y-intercepts below.

Example 2

The graphs of

y = \frac{x + 2}{x}

and

x = 3

do not cut the y-axis at any point in each graph. Thus, both of these graphs do not have a y-intercept.

In Figure 4, the behavior of the graph of $y = \frac{x + 2}{x}$ grows closer and closer to the y-axis but never touches it. This is called an asymptote. It does look like it intersects or will intersect the y-axis after some point but if we look closely at the graph, we can see that it does not touch the y-axis no matter how close it will get.
The graph of $x = 3$ is a vertical line that passes through the point $(3, 0)$ . The graph of $x = 3$ is parallel to the y-axis, thus it is not possible for this graph to cross the y-axis at any point.

In conclusion, a graph does not always necessarily have a y-intercept. Graphs that are asymptotic to the y-axis and graphs that consist of a vertical line not passing through the origin do not have y-intercepts.

Now that we have geometrically identified a y-intercept in any given graph, let’s proceed to learning how to find them even when the graph is not plotted for us.

Solving for the Y-intercept of a Given Function

Even when we have no idea what the graph of a certain function looks like, we can still determine the y-intercept of that function. Remember that one of the roles of the y-intercept is that it helps describe the graph by determining at what point the graph will intersect the y-axis.

Observing the obtained y-intercept from previous examples, we get that the y-intercept of a function is the point with the form $(0, b)$ . Thus, we can get the value of $b$ when we substitute $x$ for zero, then find the value of $y$ . Note that the graph crosses the y-axis whenever $x = 0$ . Therefore, for any given function $y = f (x)$ , the y-intercept of the function is at the point $(0, f (0))$ .

However, in cases where the function is not defined at $x = 0$ , the function has no y-intercept.

Example 3

We verify the y-intercepts we get from the previous example.

Let $y = 4 x - 6$ . When $x = 0$ , we have:
$y = 4 (0) - 6 = 0 - 6 = - 6.$

Thus, the y-intercept is the point $(0, - 6)$ .

Consider the function $f (x) = 8 - x^{2}$ . At $x = 0$ , the value of $f (0)$ is:
$\begin{array}{r} f (0) = 8 - 0^{2} = 8 - 0 = 8. \end{array}$

This means that the function has y-intercept of $(0, 8)$ .

The function $y = 1 - e^{x}$ has y-intercept at the origin, $(0, 0)$ , because when $x = 0$ , the value of the y-coordinate is:
$\begin{array}{r} y = 1 - e^{0} = 1 - 1 = 0. \end{array}$

Hence, even without the graph, we will still get the same y-intercept by substituting zero for the value of $x$ .

Example 4

Consider the rational function

f (x) = \frac{\sqrt{x + 9}}{2}

. The value of

f

x = 0

f (0) = \frac{\sqrt{0 + 9}}{2} = \frac{\sqrt{9}}{2} = \frac{3}{2} .

Thus, the function has a y-intercept at the point

(0, \frac{3}{2})

Example 5

Let

f (x) = \frac{4}{\sqrt{x - 4}}

. The function has no y-intercept because the function is not defined at

x = 0

. Take note that it is not possible for

x

to be zero because we will have

\sqrt{- 4}

in the denominator, and the square root of a negative number does not exist in the real line.

Y-intercept of Polynomial Function

In general, if we have a polynomial function of some degree $n$ ,
$f (x) = a_{n} x^{n} + a_{(} n - 1) x^{(} n - 1) + \dots + a_{2} x^{2} + a_{1} x + a_{0},$
where $a_{i}$ , for $i = 0, 1, 2, \dots, n$ are real coefficients of the polynomial, then the y-intercept of the polynomial function $f$ is the point $(0, a_{0})$ .

Example 6

Given the function $f (x) = x^{3} - 7 x^{2} + 9$ . The function is a polynomial function, thus the y-intercept of the given polynomial function is $(0, 9)$ .

How To Find Y-intercept From Two Points

In finding the y-intercept of a graph given two points in the line, we have to solve for the equation of the line in the slope-intercept form.

Note that in a linear equation of the form:
$y = m x + b,$

the slope of the line is $m$ and the y-intercept is at $(0, b)$ .

So, if we have two points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ , the slope of the line passing through these points is given by:
$m = (y_{2} - y_{1}) / (x_{2} - x_{1}) .$

After solving for the slope $m$ , we only have to find the value of $b$ . So we take one of the points, say $A (x_{1}, y_{1})$ , and substitute it for the values of $x$ and $y$ .
$y_{1} = m x_{1} + b$

Solving for $b$ , we have:
$b = y_{1} - m x_{1} .$

Then, we have the y-intercept at the point $(0, b)$ .

Example 7

Given the points

(- 2, 5)

and

(6, 9)

. First, we solve for the slope.

m = \frac{9 - 5}{6 - (- 2)} = \frac{4}{8} = \frac{1}{2} .

Thus, the slope is

m = \frac{1}{2}

. Now, we take one of the points, say

(- 2, 5)

, to solve for

b

\begin{aligned} b & = 5 - m (- 2) \\ = 5 - (\frac{1}{2}) (- 2) = 5 - (- 1) \\ = 5 + 1 = 6 \end{aligned}

We get that

b = 6

; thus, the y-intercept of the line that passes through the points

(- 2, 5)

and

(6, 9)

(0, 6)

.Note also that even if we choose the other point

(6, 9)

, we will still get the same value for

b

since both of the points lie in the same line.

Conclusion

The use of y-intercepts is deemed significant in the higher applications of linear equations and other linear models. Hence, it is important that we know how to determine the y-intercept of a function be it in a graph, in an equation format, or a linear function represented by just two points.

The y-intercept of the graph is the point where the graph of the function and the y-axis meets, and a graph that is asymptotic or parallel to the y-axis does not have a y-intercept.
The y-intercept of any given function $f (x)$ is the point $(0, f (0))$ .
The y-intercept of any polynomial function $f (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ is $(0, a_{0})$ .
A function has no y-intercept if the function is undefined at $x = 0$ .
Given two points passing through a line, the y-intercept of the line is the point $(0, b)$ , where $b = y_{1} - m x_{1}$ and $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ is the slope of the line.

In this guide, we discussed and solved for y-intercept in different mathematical scenarios, we also learned the importance of the y-intercept. Understanding how it works can help you use it better for your own benefit, such as plotting data and solving for other unknown variables; just remember that once you have the y-intercept, you can find your other variable by using a formula and plugging in what you know.

Images/mathematical drawings are created with GeoGebra.