Y Intercept
The point at which a graph intersects the y-axis is referred to as the y-intercept. It is crucial to determine the intercepts when plotting any function in the form of y = f(x).
There are two types of intercepts: the x-intercept and the y-intercept, which are points on the function’s graph where it intersects the x and y axes, respectively.
Therefore, knowing how to find these intercepts is essential. In this article, we will explore the definition, formula, and examples of the y-intercept.
What is Y Intercept?
In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y and a y-intercept is a point where the graph of a function or relation intersects with the y-axis of the coordinate system.
The Y Intercept Formula of a line Ax + By = C is,
y=\frac{C}{B}
If, the Y Intercept Formula of a line in the slope-intercept form y = mx + c is,
y=c
Where,
m = slope of the line.
c = y-intercept of the line.
If, the Y Intercept Formula of a line in the point-slope form y – b = m(x – a) is,
Y=-am+b
Where,
m is the slope of the line.
(a,b) is a point on the line.
How to find the y-intercept:
Depending on the information you have to work with, there are various methods for locating the y-intercept. The three methods for finding the y-intercept on a graph, in a table, or using an equation are as follows:
Find the y-intercept of a linear function using the slope and a given point
Determine the graph’s slope and a point first. Write a linear equation in slope-intercept form (y = mx + b) once this is finished. Rewrite the equation using the supplied point (x, y) and slope (m), inserting the correct values for x, y, and m. To find the y-intercept using this information, calculate the equation for b.
Example:
Suppose a graph includes the point (1, -4) with a slope of -2.
Solution:
To write the linear equation in slope-intercept form, we use y = mx + b.
Substituting the given values, we get:
-4 = (-2)(1) + b
Simplifying the equation, we get:
-4 = -2 + b
Adding 2 on both sides, we get:
-2 = b
Therefore, the y-intercept is -2, which is the value of the y-coordinate when x = 0 in the equation y = -2x – 2.
Find the y-intercept of a linear function using two points from a table or graph
Find the two points in a graph or table that are displayed. Take note of each point’s coordinates (x, y) first. Find the rise and run to determine the slope using this information. Find the difference between the y-coordinates of the two places to determine the rise. Find the difference between these two places’ x-coordinates to get the run’s length. To calculate the slope, divide the difference in the y-coordinates by the difference in the x-coordinates.
Write a linear equation in slope-intercept form (y = mx + b) once the slope has been determined. Replace the correct values for x, y, and m in the equation using one set of coordinates (x, y), and the slope, m. After that, find the y-intercept by solving the equation for b.
Y-intercept Example
Suppose a graph contains the points (3, 6) and (-1, -2). Find the y-intercept.
Solution:
Here are the steps to find the y-intercept of the line passing through the points (3, 6) and (-1, -2):
| Step | Calculation | Result |
| 1 | Rise = 6 – (-2) = 8 |
| Run = 3 – (-1) = 4 | ||
| Slope = Rise/Run = 8/4 = 2 | 2 | |
| 2 | Substitute slope and one point into slope-intercept form | |
| y = mx + b | ||
| 6 = 2(3) + b | ||
| 6 = 6 + b | ||
| b = 0 | 0 |
Therefore, the equation of the line passing through the points (3, 6) and (-1, -2) is y = 2x + 0 or simply y = 2x. The y-intercept is 0.
Find the y-intercept of a linear function using an equation
If you already know the line’s equation, use algebraic calculations to determine the y-intercept. Replace x with 0 in the equation’s relevant position since the y-intercept always corresponds to an x-value of 0. Then, solve. If you already know the line’s equation, use algebraic calculations to determine the y-intercept. Replace x with 0 in the equation’s relevant position since the y-intercept always corresponds to an x-value of 0. Then, solve.
Example:
Find the y-intercept of the line 4x + (-3y) = 24.
To find the y-intercept,
let x = 0. 4(0) + (-3y)
= 24 -3y
= 24 y
= -8
Therefore, the y-intercept is -8.
Finding the y-intercept in a quadratic function
The y-intercept of a quadratic function is the location where the parabola crosses the y-axis. The y-intercept in the shown graph is -3.
When the equation of a line is written in slope-intercept form (y=mx+b), the y-intercept is the constant term, which is represented by the variable b. For instance, in the linear equation y=3x+2, the y-intercept is 2.
The point where a line’s graph crosses the y-axis is known as the y-intercept. Given that the graph crosses 4 on the y-axis, the y-intercept in the coordinate plane is 4.
The y-intercept is significant because it indicates what y will be when x = 0. It gives a linear function a place to start.
The line’s intersection with the y-axis on a graph is known as the y-intercept. In every case, the corresponding x-coordinate is 0. By multiplying rise over run, one may get the slope. This is accomplished by calculating the difference between the y- and x-coordinates and then dividing this difference.
When a linear equation is expressed in the slope-intercept form (y=mx+b), the variable m stands in for the slope. It is the equation’s coefficient relating to x. The variable b stands in for the constant y-intercept.
The constant variable b stands in for the y-intercept when a linear equation is expressed in slope-intercept form (y=mx+b). For instance, the variable b corresponds to 8 in the equation y=6x+8. The y-intercept is shown here.
The yy-intercept is the yy-value that corresponds to xx when x=0x=0. In real life, this often refers to the starting point when something is being measured.
For instance, consider population change in the United States.
In this scenario, the xx-values could represent time, measured in years. The yy-values could represent the population, measured in millions of people. When x=0x=0, this value represents the starting year for measuring population change. The corresponding yy-value represents the size of the population in the starting year. This value is the yy-intercept.
