4 Ways How to Find a Y-Intercept?

Finding Y-intercepts is not as complicated as it looks, intercepts are simple in themselves. They are simple points where the graph of the equation crosses the Y-axis or where the value of x=o. In this article, learn how to find the y-intercept or vertical intercept of a function and graphs in detail with examples

When finding the Y-Intercept of a graph or table you look for the point of intersection between the equation and the Y-axis. For a straight line, it occurs only once but for a graph it does not have a straight line, there can be different y-intercepts in the same graph.  

The vertical intercept is the same as the Y-intercept because most graphs and equations set Y as the vertical line on the graph. On an (x,y) coordinate graph the x coordinates from the horizontal and y coordinates from the vertical line, where a line between the coordinates crossing the vertical line is Y intercept. 

There is up to one way to find the y-intercept but it depends on your starting information. There are three ways below to determine the y-intercept on a graph, in a table, or with an equation.

Finding Y intercept Using Slope and a Given Point 

First of all, identify the slope and a point on a graph, then write a linear equation in slope-intercept form (y=mx+b). Now rewrite the equation using given points (x, y) by replacing the suitable values for x, y, and m. for example, 

Just consider that a graph contains the point (-3, 5) where the slope is equal to 3.

Write a linear equation in slope-intercept form like that,

Y = mx + b

Now, substitute the corresponding value of the variables, 

5 = (3)(-3) + b

Solve one-step equation from here,

5 = -9 + b

5 + 9 = b

14 = b

So, the y-intercept is 14. 

Finding Y-intercept Using two Points from a Table or Graph

First of all, determine two coordinates (x, y) using a table or a graph and find the rise and run to identify the slope. Now find the difference in y-coordinates of the two points and calculate the rise. Then find the difference in the x-coordinates of the same two points and calculate the run. Now find the slope by dividing the difference of y-coordinates by the difference of x-coordinates, then solve the equation from the above method.

For example, a graph consists of two points (2, 4) and (-2, -4) to find the value of b.

Calculate the rise and run to find the slope which is 2.

 rise/run = 4-(-4)/ 2-(-2) = 8/4 = 2

Now, write the equation in slope-intercept form, similar to the above method.

Y = mx + b

4 = (2)(2) + b

4 = 4 + b

4 – 4 = b

0 = b

Hence, b = 0, and the y-intercept is 0.

Finding y-intercept Using an Equation

When you already have an equation of the line, you should solve it algebraically to find the y intercept that already has a corresponding x-value of 0, substitute x with 0, and solve the equation.

For example the equation of the line is 4x + (-5y) = 10

By replacing the value of x.

4(0) + (-5y) = 10

0 – 5y = 10

Y = 10/-5

Y = -2

So, the y intercept is -2 of the line equation.

Finding Y-intercept in Quadratic Function

Finding the y-intercept of a quadratic function or parabola is the same as that of a line, where the parabola crosses the y-axis. We can write the standard form of a quadratic function as 

Y = ax² + bx + c, where x & y are variables and a, b, c are constants. Find y intercept by replacing 0 as the value of x as given below.

For example, the quadratic equation is y = 4x² + 2x +2

Y = 4x² + 2x + 2

Y = 4(0) + 2(0) + 2

Y = 0 + 0 + 2

Y = 2

So the y-intercept is 2 of the quadratic equation.