How to Find x and y Intercepts Of Graphs?
How to find the x and the y intercepts of graphs of functions and equations?
The x and y intercepts of a graph are points of intersection of the graph with the x axis and the y axis respectively. This is a tutorial with examples and detailed solutions on how to find these points.
Example 1
Find the x and the y intercepts of the graph of function f defined by
Solution to Example 1

Since a point on the y axis has x coordinate equal to zero, to find the y interecpt, we set x to zero and find the y coordinate which is f(0).
f(0) = 3(0) + 9 = 9

A point on the x axis has y coordinate equal to 0, to find the x intercept, we set y = f(x) = 0 and solve for x
f(x) = 3 x + 9 = 0

Solve for x.
x = 3

The x and y intercepts of the graph of f are
x intercept: (3 , 0)
y intercept: (0 , 9)
Example 2
Find the x and the y intercepts of the graph of the equation the circle given by
Solution to Example 2

To find y intercept: Set x = 0 in the equation and solve for y.
(0  1)^{ 2} + (y  2)^{ 2} = 16

Solve for y
1 + (y  2)^{ 2} = 16
(y  2)^{ 2} = 15
solutions: y_{1} = 2 + √(15) and y_{2} = 2  √(15)

To find x intercept: set y = 0 in the given equation and solve for x
(x  1)^{ 2} + (0  2)^{ 2} = 16
solutions: x_{1} = 1 + √(12) and x_{2} = 1  √(12)

The x and y intercepts of the graph of the given equation are
x intercepts: A = (1  √(12) , 0) and B = (1 + √(12) , 0)
y intercepts: C = (0 , 2  √(15)) and D = (0 , 2 + √(15))

The graph shown below is that of the given equation. Examine the x and y intercepts and compare to those calculated. Note that the x and y intercepts may be determined graphically.
Example 3
Calculate the x and the y intercepts of the graph of the linear equation given by
Solution to Example 3

Set x = 0 in the given equation and find the y intercept.
3(0) + 2y = 6

Solve for y
y = 3

Set y = 0 and solve for x to find the x intercept
3 x + 2(0) = 6 , x = 2

The x and y intercepts of the graph of the above equation are:
x intercepts: A = (2 , 0)
y intercepts: B = (0 , 3)

The graph of the given equation is shown below. The x and y intercepts are those calculated above. Note that the x and y intercepts may be determined graphically.
Example 4
Calculate the x and the y intercepts of the graph of the quadratic function given by
Solution to Example 4

Set x = 0 in the formula of the given function and calculate the y intercept which is equal to f(0).
y = f(0) = 3

To find the x intercept: set y = f(x) = 0 and solve for x
 x^{2} + 2 x + 3 = 0
Solutions: x_{1} = 1 and x_{2} = 3

The x and y intercepts of the graph of the above equation are:
x intercepts: A = (1 , 0) and B = ( 3 , 0)
y intercepts: C = (0 , 3)

The graph of the given function is shown below along with the x and y intercepts as calculated above.
Example 5
Determine the x and the y intercepts of the graph of the logarithmic function given by
Solution to Example 5

Set x = 0 in the formula of the function and the y intercept is equal to f(0).
y = f(0) =  ln(0 + 1)  2 =  2

Set y = f(x) = 0 and solve for x
 ln(x + 1)  2 = 0
ln(x + 1) = 2
x + 1 = e^{2}
solution: x =  1 + 1/e^{2}

The x and y intercepts of the graph of the above equation are:
x intercepts: A = (1+1/e^{2} , 0)
y intercepts: B = ( 0 ,  2)

The graph of the given function is shown below along with the x and y intercepts as calculated above.
Example 6
Calculate the x and the y intercepts of the graph of the exponential function given by
Solution to Example 6

The y intercept is equal to f(0).
y = f(0) = e^{0 + 1}  2 = e  2

Set y = f(x) = 0 and solve for x
e^{x + 1}  2 = 0
e^{x + 1} = 2
x + 1 = ln 2
solution: x =  1 + ln 2

The x and y intercepts of the graph of the above equation are:
x intercepts: A = (1 + ln 2 , 0)
y intercepts: B = ( 0 , e  2)

The graph of the given function and its x and y intercepts are shown below.
Example 7
Calculate the x and the y intercepts of the graph of the rational function given by
Solution to Example 7

The y intercept is equal to f(0).
y = f(0) = 2/3

Set the numerator of f(x) equal to zero and solve for x to find the x intercepts
x^{ 2}  x  2 = 0
solution: x_{1} =  1 and x_{2} = 2

The x and y intercepts of the graph of the above function are:
x intercepts: A = (1 , 0) and B = (2 , 0)
y intercepts: C = ( 0 , 2/3)

The graph of the given function and the x and y intercepts are shown below.
Example 8
Calculate the x and the y intercepts of the graph of the sine function given by
Solution to Example 8

The y intercept is equal to f(0).
y = f(0) = 1/2

Set f(x) equal to zero and solve for x to fnd the x intercepts
sin(x) + 1/2 = 0 , sin(x) = 1/2
solution:Because of the periodicity of the sine function, there is an infinite number of x intercepts given by:
x_{1} = 7π/6 + 2kπ , k=0,~+mn~1 , ~+mn~2 , ...
x_{2} = 11π/6 + 2kπ , k=0,~+mn~1 , ~+mn~2 , ...

Some of the x intercepts and the y intercept are:
x intercepts: A = (π/6 , 0) , B = (7π/6 , 0) and C = (11π/6 , 0)
y intercepts: D = ( 0 , 1/2)

The graph of the given function and the x and y intercepts are shown below.