What Is the Y-intercept?
The y-intercept is the value of the y-coordinate of the point where a graph intersects the y-axis. So, the x-coordinate of the point corresponding to the y-intercept is zero.
In the image, the graph crosses the y-axis at the point (0,-2), so its y-intercept is -2.
[Note: Here, we will take linear equations as the function.]
How Do We Find the Y Intercept?
We can find the y-intercept of a linear equation using three different methods, which are:
Method 1: Using slope and a given point
Step 1: Find the slope and the coordinates of a point that lies on the graph of the function.
Step 2: Write the slope-intercept form of the linear equation and substitute the values of the coordinates, x and y and the slope, m.
Step 3: Solve for y-intercept, b.
Method 2: Using two points
Step 1: Find two points that lie on the graph of the function using a graph or table.
Step 2: Calculate slope using the formula:
\(Slop, m = \frac{Rise}{Run} = \frac{Change\text{ }in\text{ }y\text{ }coordinates }{Change\text{ }in\text{ }x\text{ }coordinates}= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Step 3: Write the linear equation in slope intercept form: y = mx + b.
Step 4: Substitute the coordinates of one point (x,y) and slope, m, in the above equation and solve for y-intercept, b.
Method 3: Using an equation
Step 1: Write the given equation of the line.
Step 2: Substitute ‘x = 0’.
Step 3: Solve the equation for y. The value of y is the y-intercept.
Rapid Recall
Solved Examples
Example 1: Find the y intercept.
a. Line with a slope of 2 and passing through (2, 3) on the coordinate plane.
b. Line passing through (3, 1) and (-1, 4) on the coordinate plane.
c. Line 3y = -2x + 2
Solution:
a. Line with a slope of 2 and passing through (2,3) on the coordinate plane.
y = mx + b [Write slope intercept form of linear equation]
3 = 2 (2) + b [Substitute values]
3 = 4 + b [Simplify]
b = 3 -4 [Solve for b]
b = -1 [Simplify]
So, the y-intercept of the line is -1.
b. Line passing through (3, 1) and (-1, 4) on the coordinate plane.
\(Slop, m = \frac{Rise}{Run} = \frac{Change\text{ }in\text{ }y\text{ }coordinates}{Change\text{ }in\text{ }x\text{ }coordinates} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) [Formula for slope]
\(= \frac{4 – 1}{-1 -3}\) [Substitute values]
\(= – \frac{3}{4}\) [Simplify]
\(y = mx + b\) [Write slope intercept form of linear equation]
\(1 = \left( -\frac{3}{4} \right)\left( 3 \right) + b\) [Substitute values]
\(1 = -\frac{9}{4} + b\) [Simplify]
\(b = 1 + \frac{9}{4}\) [Solve for b]
\(b = \frac{13}{4}\) [Simplify]
The y-intercept of the line is \( \frac{13}{4}.\)
c. Line 3y = -2x + 2
\(3y = – 2x + 2\) [Rewrite equation]
\(3y = – 2\left( 0 \right) + 2\) [Substitute x = 0]
\(3y = 0 + 2\) [Simplify]
\(3y = 2\) [Simplify further]
\(y = \frac{2}{3}\) [Solve for y]
The y-intercept of the line is \(\frac{2}{3}.\)
Example 2: The cost of entering a fest and taking a ride on a roller coaster (y) in dollars and the number of rides on the roller coaster, x are related by the equation 2y = 8x + 10. Find and interpret the y-intercept.
Solution:
\(2y = 8x + 10\) [Rewrite equation]
\(2y = 8\left( 0 \right)+ 10\) [Substitute x = 0]
\(2y = 0 + 10\) [Simplify]
\(2y = 10\) [Simplify further]
\(y = \frac{10}{2}\) [Solve for y]
\(y = 5\) [Simplify]
The y-intercept is 5. So, the entry fee or the cost of entering the fest is $5.
The x-coordinate of the point where a graph intersects or crosses the x-axis is called the x-intercept.
Slope measures the steepness of a line. Mathematically, it is equal to the value of the ratio of rise to run between two points that lie on the line.
The slope intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept.
An ordered pair represents the x and y coordinates of a point on the coordinate plane.
A line passing through origin (0,0) has a y intercept of 0.