Intercepts are the segment lengths of the graphs that cut off the Coordinate Axes. Intercepts are important for sketching the graphs of a functions. The measure and number of x intercepts in a graph also hint at the turning points in the graph. The points where the graphs cut the axes are also often mentioned as intercepts.

Intercept

The graph of the straight line 4x + 3y = 12 is seen above. You may notice the graphs cuts off  x- intercept measuring 3 units and y - intercept measuring 4 units each. The corresponding intercept points (3, 0) and (0, 4) are also marked.

In application models, the intercepts play a significant role. For example, the y intercept of a cost function refers to the fixed cost and the x intercepts of the Marginal cost function refers to optimal output required to minimize the cost.

The x - intercept is measured along the x - axis. The y - coordinate of the point representing the x intercept as zero. This means the x - intercept can be found by setting y = 0 or solving the equation f(x) = 0.

Solved Examples

Question 1: Find the x intercept of the graph of  2x - 3y = 12.
Solution:
 
2x - 3(0)  = 12                           Set y = 0
2x = 12          
x = 6                                       Solved for x.
Hence the straight line 2x - 3y = 12 cuts off a x - intercept = 6  or intersects the x axis at the point (6, 0).

Y - intercept of a graph is the length measured along the y axis or y coordinate of the point representing the y - intercept. So to find the y intercept, we set x = 0 and solve for y, or we evaluate f(0).
 

Question 2: Find the y - intercept of the graph of 2x - 3y = 12
Solution:
 
2(0) - 3y = 12                            Set x = 0
      - 3y = 12
         y = -4                              Solved for y.
Hence the straight line cuts off a y intercept = -4 or intersect the y axis at (0 , -4).
 

Question 3: Let us find the x and y intercepts of the quadratic function f(x) = x2 - 9x + 14.
Solution:
 
At the x intercept y = 0 or f(x) = 0
x2 - 9x + 14 = 0
(x - 7)(x - 2) = 0                         Equation is written in factored form
⇒ x - 7 = 0   or x - 2 = 0             Zero factor property
x = 7 or x = 2

Hence the x - intercepts are 2 and 7 and the points representing the x intercepts are (2, 0) and (7, 0).
The y - intercept is given by f(0).

f(0) = 02 - 9(0) + 14 = 14
Thus the y - intercept is 14 or (0, 14)
The graph of the function is given showing the intercepts.

Quadratic Graph
 

The slope intercept form of a linear equation can be considered as a formula to find the y - intercept of the linear graph.
y = mx + b is the slope intercept form of a a linear equation where m represents the slope of the line and the b is the y - intercept of the graph.This form of linear equation is used for making a table of ordered pairs to be plotted while graphing the linear function. This form of linear equation is also useful for finding the y - intercept direct by observation without having to do any algebraic treatment.

For example, for the line represented by the equation y = 3x - 7, the y intercept is -7.
The intercept form of a linear equation, is written using the x and y intercepts as follows:

$\frac{x}{a}$ + $\frac{y}{b}$ = 1 where a and b are correspondingly the x and y intercepts of the line.

Intercept Form

Writing a linear equation in Intercept form provides a method the find the x and y intercepts of the equation.

Solved Example

Question: Find the x and y intercepts of the line  5x + 7y = 35.
Solution:
 
To write the equation in Intercept form we require 1 on the right side of the equation.

$\frac{5x}{35}$ + $\frac{7y}{35}$ = $\frac{35}{35}$                                          Divide the equation by 35.

$\frac{x}{7}$ + $\frac{y}{5}$ = 1                                                         Equation is written in intercept form.

Hence the x and y intercepts are correspondingly 7 and 5.

The two intercepts (a, 0) and (0, b) are sufficient to draw the graph of a straight line. In the above example, line 5x + 7y = 35 can be graphed by plotting the points (7, 0) and (0, 5) and joining them.